On the surface the claim in the question is very plain. According to CBS News, the chief of Russia's space agency Dmitry Rogozin stated that because of the more successful track record of the Soviet Union with regard to the exploration of Venus it was "a fact that the United States called Venus 'the Soviet planet'" (the Russian quote is "США называли Венеру «советской планетой»"). The question is: Is this really a fact?
There's one thing to get off the table first. "Soviet planet" is not about legal claims on Venus. Rogozin's quote is clearly about the scientific track record. The Soviet Union launched many successful missions to Venus from the 1960s to the 1980s. It is mostly accepted that their Venus exploration program was more successfully than the NASA counterpart. So it makes sense somehow to describe Venus as the "Soviet planet". But is it true what the quote states as a fact? Did the United States (meaning a large part of the U.S. public) really call Venus "the Soviet planet"?
I have to admit that my answer is far too long for this question that perhaps has only little significance. I present this detailed answer because I want to illustrate that sometimes answering a question on this site needs more effort than just some web searches and a superficial look at the results. Sometimes much more care is needed to compose an answer that really adheres to the scientific method. But this is the standard for good answers on this site.
Here's the TL;DR: if you don't want to go through everything.
I couldn't find an instance in which a U.S. writer called Venus the "Soviet planet" or "Russian planet". All in all the number of instances is small for the English language. The word is not common at all in English. There is no evidence that it was at any point in time. The few instances that I found were either from Russian writers writing mostly for Russian readers or from British writers. Based on this there isn't much support for the claim that "the United States called Venus 'the Soviet planet'".
Which evidence do you look for?
First I'd like to discuss what type of evidence is needed in order to falsify or support the claim made by Rogozin. In the best case a good answer on this site would use the scientific texts that investigate the words used by American speakers to call the planet Venus in the light of the Soviet Union space exploration program. Unfortunately these texts don't exist.
In cases like this many answers on this site search the internet for direct quotes. Next the quotes are used as evidence either in favor or against the claim. Actually this may be considered original research but it seems to acceptable on this site. In this spirit you could conduct a web search (perhaps augmented by other databases). If that search returns a large-ish number of instances that are produced by writers from the U.S. and which contain the word, this would be sufficient evidence to support the claim.
But what counts as a "large-ish number of instances"? Is a single quote that can be linked to a U.S. writer enough for supporting the claim? Are three quotes enough? How many do you need? And what about falsification? Which evidence is enough to falsify the claim? Under which circumstances would you conclude that it's not true that "the United States call Venus 'the Soviet planet'"? What if there is not a single use of the word that can be unambiguously linked to a U.S. writer? What if there is only a small number of instances that are all debatable?
These questions illustrate that although the OP question seems rather plain it may still be very hard to answer or perhaps even unanswerable because of the limits that this site imposes on answers with regard to original research. At minimum any attempt to answer must be very careful in the interpretation of its evidence (especially if the data set it is based on is rather small). An answer that cherrypicks only few examples but claims that they show a general presence the word in the U.S.A. would need to defend this claim.
This answer tries to take all these concerns into consideration. It uses a transparent methodology and lays out the data clearly and (hopefully) unbiased. It doesn't rely on a single data source but tries to approach the question from several angles. Five different search strategies will be used to collect instances in which Venus is called the "Soviet planet"/"Russian planet":
- A web search restricted to official U.S. documents
- A Google Scholar search that looks at academic texts (articles and books)
- A Google Books search that includes also non-academic books and magazines
- A web search restricted at documents older than 2020
- A search of linguistic corpora
I investigated all instances (where that was possible) that each strategy returned. For each instance I checked the context of the utterance and tried to identify the origin of the writer. Note that the Google searches that I use are not necessarily reproducible. As far as I know Google adjusts search results based on location and search history so I wouldn't be surprised if your search results differ from mine. I still hope that my methodology is sufficiently reliable so that these differences do not falsify my answer.
Official U.S .documents
The first line of evidence comes from a Google search restricted to web servers run the U.S. government (which includes NASA servers). Many of these servers host scans of official documents going back to the 1960s. If Venus was often called the "Soviet planet" by NASA or other branches of the U.S. government it's reasonable to assume that this search will return at least some documents.
However this is not the case. The search returns a single document hosted on a C.I.A. server. The document is a collection of library cards that contain abstracts of space-flight related text from the U.S.S.R. in the 1960s. The card on p. 1 contains "Soviet planet" and the (unrelated) card on p. 3 contains the word "Venus". Thus there is no evidence that Venus was called either the "Soviet planet" or the "Russian planet" in U.S. government documents.
Another way of collecting evidence for a common usage of the word is through the scientific literature. Google Scholar is a search engine for academic documents. It also searches documents that are behind a paywall and therefore may be missed by a normal search engine.
Google Scholar query
The search on Google Scholar returns fourteen matches in total. Two matches come from the 1991 book "Living Earth: A short history of life and its home" by the British geologist Euan George Nisbet. One match is incorrectly attributed to an "RE Nisbit" by Google Scholar. This book repeats the OP claim, as Nisbet writes on p. 8:
There is rivalry in science, just as everywhere else, and Venus had become known as the Russian planet, whereas Mars was an all-American affair.
Another match is called "Lavochkin and Kovtunenko scientific and production association: Intersection of destinies that gave a new impetus to the development of national unmanned astronautics". This is a 2012 article by I.L. Shevalev. The article was first printed in Russian in a Russian journal before translated into English. The writer clearly identifies as Russian when he writes on p. 581:
Due to the intensity and impact of visits to Venus by the spacecraft "Venera", the world community even called it "the Soviet planet". Only in the period 1978–1985 did we send to Venus eight space vehicles (four missions). For comparison, the USA, as one of the most important space powers, did not sent any spacecraft.
Google scholar also lists a document called "КОСМИЧЕСКАЯ ДИПЛОМАТИЯ В XXI ВЕКЕ (ПОДХОДЫ РОССИИ, ИНДИИ И ЕВРОПЕЙСКОГО СОЮЗА К ИСПОЛЬЗОВАНИЮ" УМНОЙ СИЛЫ")" which is the Master's thesis of a student of International Relations at St Petersburg University. Without stating their source the writer says on p. 39:
The mission that is planned to involve both orbiter and lander and the aim is to conduct remote-sensing observations by radars, might be perceived as a Russian return to the space-interest of USSR when Venus was sometimes called “Russian planet”, as vast amount of Soviet Venera program spacecrafts was launched to analyze Venus's surface and atmosphere.
The C.I.A. library card that was found in the U.S. government search makes a return in another Google Scholar document called "DA Pamphlet 70-5-8, USSR: Missiles, Rockets, and Space Effort, a Bibliographic Record, 1956-1960, 28 September 1960". On p. 23 you find the following instance (which is the same as in the C.I.A. document). It doesn't refer to Venus, and it is older than the Soviet missions to Venus:
SOVIET PLANET. Sovetskaia planeta, by I. Merkulov, in Kryl'ia Rodiny (Mar 1959) 8-9. In Russian. Description of the Soviet cosmic rocket, including its trajectory, instrumentation, and launching.
Finally there is "Text analysis of Maxpark and LiveJournal Russia: How is the evaluation of modern femininity and masculinity discussed in Russian blogs". This is a Master's thesis in Modern Languages and Cultural Studies at the University of Alberta, which mentions on p. 102 a blog hosted in Russia named "Russian Planet" and the deity Venus on p. 66.
Based on the Google Scholar search results "Soviet"/"Russian planet" refers to the planet Venus in one text written by a British geologist and two texts either written in Russian or by Russian writers mostly for Russian readers. U.S. writers were not found in the search.
Books and non-academic texts
Some hits returned by Google Scholar are Google Book links. I searched this database for "Soviet planet"/"Russian planet" in combination with "Venus". Google Books contains also non-academic texts. By default they are sorted by "Relevance".
Google Books query
Unfortunately Google Books is a fairly messy thing. Superficially the number of hits seems very high. But if you look at some books appearing on a later sub-page, "Soviet planet"/"Russian planet" is not contained in the book. It's not possible to look into all the books returned by the search. Therefore I investigated only the books that came with a text snippet with the search words "Soviet planet"/"Russian planet" (I excluded phrases like "competent Soviet planet flights") as well as "Venus". This method produced ten titles on the first two sub-pages. I didn't find any more titles with matching snippets on the first five sub-pages.
The first title is "The scientific exploration of Venus" by Fredric W. Taylor. He writes on p. 64 about the scientific space program of the Soviet Union:
One reason, of course, was that the greatest champion of Venus exploration, the Soviet Union, underwent a major political upheaval, one consequence of which was that their scientific space programme became moribund. Where they had once regarded Venus as a "Russian planet" that showcased their skills and outlook, much as manned landings on the Moon had done for the United States, only a few scientists remained engaged in Venus research
Later on p. 244 Taylor (who is another British scholar) writes:
There is still something of a feeling, and not just in Russia itself, that Venus is a "Russian planet" where space exploration is concerned. This reputation was well earned by the ambitious and successful programme that was mounted in the 1960s, 1970s and 1980s with the Venera and Vega missions.
The second book is Nisbet's "Living Earth" which was discussed earlier. Next is "Lives of the Planets: A natural history of the solar system" by the British scientist Richard Corfield who says on p. 112:
If Venus is the "Russian planet", then Mars is, without doubt, the American one.
Two titles are from different volumes of the Foreign Broadcast Information Service/Joint Publications Research Service "USSR Report: Space". Similar to the C.I.A. library cards mentioned earlier they are a survey of the space-exploration texts from the Soviet Union with English translations. "Soviet planet" can be found in the 1990 report on p. 29 in the translation of a 1989 U.S.S.R. article:
Our research is limited for the time being to the "Soviet planet", Venus, where we have had a lot of success, and Mars (where there are more failures).
The 1985 report contains on p. 125 another reference to the word in a U.S.S.R. text:
And the further successes of Soviet unmanned spacecraft created in Babakin's design bureau even after him were so impressive that thanks to the scientific results obtained as a whole Venus began to be called the "Russian" planet.
The Google Books snippet shown for "Books for Children" is weird:
The experiences and reactions tric uncle on a space voyage to Nede , satellite of the young people and their elusion of Russian planet to Venus . The fascinating , vividly de patrols with the help of Kai ' s long - lost , amnescribed planet is ...
An investigation of this quote shows that the OCR mixed up words from two separate columns. The corrected passage is about "a space voyage to Nede, satelite planet to Venus" and also about "young people and their elusion of Russian patrols".
The 1995 book "Exploring the Solar System" by British writer Nicholas Booth has the following passage on p. 85:
But whereas the Venera series of probes had made Venus the "Russian planet", fate conspired to make Mars very much "US territory". Good opportunities to launch spacecraft to Mars occur every 25 months, when the relative positions of the Earth and Mars are such that a spacecraft can coast between the two using reasonably limited amounts of fuel.
The remaining three quotes are from 1959 and 1960. They are false positives because they refer to the Sputnik satellite when they use "Russian planet". The first is from Newsweek, Volume 53 1959, p. 23. It contains the passage "Lost in the excitement over the Russian planet was a new Atlas transmission triumph". The quote from the Journal of the British Interplanetary Society, Volume 17 refers to a 1959 New Scientist article "The Russian planet and its instruments". The third is from vol. 11 of Meteorological and Geoastrophysical Abstracts. It describes a Dutch text "De eerste kunstmatige planeet" (in English: "The first artificial planet") from 1959. The summary contains the sentence "A fig. shows the track of the artificial planet in the earth–moon system, and another figure shows the paths of Venus, Earth, Mars and the Russian planet." These quotes are older than the Soviet Venus missions and so the word can't refer to that planet.
Searching the web
A normal web search is not very helpful because the statement containing the current claim was covered by many news websites. Therefore I restricted my search to websites that Google indexes as older than Dec. 31 2019. I also excluded the number 2020 from the web results.
This search returns only 15 pages. Eleven pages are dynamic news feeds or news websites mentioned the current "Russian planet" story at the time Google indexed them. Two are from Pinterest. One is a picture with the capture "Venera 10/Soviet Planet Venus Landing Venera-10 landed on October 23, 1975". The other is a "Detailed Physical Map of Venus in Russian". One hit is a PDF copy of "The scientific exploration of Venus" discussed above. The remaining hit is from the 2017 blog entry "Kaluga: the cradle of cosmonautics" that discusses the Russian Venus missions:
«Venera» is a series of Soviet automatic interplanetary stations for studying the planet Venus and its outer space. The severe conditions on Venus and the initial lack of reliable information about its surface complicated greatly the process of exploring the planet. The vehicles of the first series even could float. Their first missions failed, bur later the USSR got so much success in the study of Venus, that it was called the «Russian Planet».
The blog writers are from Saint Petersburg.
Corpora of English
Finally I checked different corpora of (American) English. The search engines for these resources are n-gram-based. They aren't document-based like the usual Google tools. This means that I searched for "Soviet planet"/"Russian planet" but my searches didn't look for "Venus" in the same text. Still this means that the searches return a superset of the instances where the word refers to Venus.
The Corpus of Contemporary American English is a database containing a billion words from news, magazines, books, academic texts, t.v. shows etc. from 1990 to 2019. Despite being one of the largest corpora of the English language "Soviet planet"/"Russian planet" is not found in the database:
The word also doesn't occur in the Corpus of Historical American English, a 400 million words corpus with texts from 1810 to 2000:
The 14 billion Word Web Corpus has two instances of "Russian planet", but judging by the context the instances do not refer to Venus:
I also had a look at the Google Ngram Viewer. This tool uses the Google Books database to show the distribution of occurrences during the select time span. The number of instances is so small that the tool warns that the searches returned only a single result.
Google Ngram query
Here's the resulting graph:
There are peaks around the year 1960. Again these peaks can't refer to the successful Soviet Venus missions which were launched a year later. The explanation for this peak becomes clear if "artificial planet" is added to the graph. We've learned above that this word referred to the Sputnik satellite. The new graph puts the peaks for "Soviet planet"/"Russian planet" into perspective:
Even without Venus "Soviet planet"/"Russian planet" are very rare in English-language books.
What does all this mean?
There are no official U.S. government documents available on the web in which Venus is called the "Soviet planet" or "Russian planet". There are three academic texts that contain the word with this meaning. One is by a British writer and two are by Russian writer written mostly for Russian readers. Furthermore the word occurs in five more books. Three books are written by British writer and two are English translations from Russian texts. A web search for instances that are older than 2020 returns a single blog post written by Russian writers. "Soviet planet"/"Russian planet" together with "Venus" can't be found even in the largest available English corpora.
This means means that there is no evidence that writers from the U.S.A. called Venus the "Soviet planet" or "Russian planet". The search produced nine instances in total. Four instances are by British writers and five instances are by Russian writers.
Of course the absence of evidence is not the same as the evidence of absence. It's possible that the word was used very often in spoken but not in written American English. It's also possible that it was used often in some written genres that were not preserved in electronic form. But why would that be the case? If Venus was perceived as "the Soviet planet" by a large part of the U.S. public why is there no written evidence? Why are there some instances by British writers only? Why should British writers use a genre that was better preserved than the genres that the American writers used?
But I'm not only surprised by the total absence of instances from American writers. I'm also surprised that most instances that my search produced were from Russian-speaking writers. With the context of the OP quote it seems that even today the idea of Venus as "the Russian planet" is very present in the mind of the Russian space exploration community. Perhaps much more so than in the mind of the English-speaking counterpart. This makes sense if you think about the turns of the Space Race in the 1960s. The Soviet Union had a head start with the early exploration of the Earth's orbit. The early Venus missions were also a huge success. Even later when the Soviet space program lost ground to the NASA missions (which even brought American astronauts to the Moon) Venus was always the single area of exploration where the Soviet effort was not shadowed by NASA.
Based on this I can imagine that the Venus missions may be a special point of pride in the Russian perspective of space exploration. From this pride it may only be a small step to assume that the rest of the world and the U.S.A. could not help but acknowledge these scientific successes. I don't speak Russian so I can't investigate how often "советской планетой" occurs in Russian sources. But based the results in this answer I guess that there are many more Russian instances of the word than I found for English.
This may explains why there are rather many Russian-based instances. What could be the reason that there are instances from British writers but not from American writers? Perhaps the outcome of the Space Race may also give this explanation. From an American Perspective using "Soviet planet" would indeed mean handing the Soviet Union a stage victory in the Race (something that may have been avoided in written language). For a British writer who comes from a country that only had stakes in the Space Race by political allegiance the word may have been less loaded than for an American writer. In this context it's interesting that each of the four instance by British writers balance the Soviet stage victory by referring to the successful American exploration of Mars or the Moon. This is a pattern that is found in only one Russian-based text.
This all adds up to the following hypothesis. The concept of "Venus the Soviet planet" is mostly a Soviet/Russian construct. Even if the English-speaking world acknowledged the scientific successes of the Soviet Venus missions, their perspective was balanced and contrasted by the NASA successes in other areas. In the light of these successes Venus may just not have been very prominent in the mind of the American public. So there would have been little reason to talk about that planet and much less to hand it over verbally to the U.S.S.R. This may be why no instance of "Soviet planet" from American writers were found in the present data.