Neither entirely right nor entirely wrong. Depends on perspective, interpretation; and source material given out by Cruz over that 'platform' is too short to definitively answer that question.
But what can be said:
This looks like extreme spin doctoring in a continued mud slinging battle over details of partisan politics: it is about House Resolution 183 which was focussed initially on antisemitism alone, and thus targetting Omar specifically, without mentioning her by name like House Resolution 241 does (or tries to). But HR183 was prompted by remarks from Omar that were widely criticised as being again antisemitic.
All Democrats then rallied to oppose this in its early draft form, passing this as it came in its "All -isms matter" version. Reps then interpreted this as not really true to intent of original idea, introduced another HR and a Senate resolution with 'pure focus'. Unless Ted knows something else, this is pure semantics along party lines and in terms of final voting on the House resolution not entirely accurate, if not plainly false.
While it may seem obvious that Omar would not have been in favour of any resolution condemning her specifically or her views, in the end of this process to pass a resolution, Omar voted in favour of that final version and she did not speak up in either way during the official debates.
Thus Mr Cruz is wrong if a short Twitter statement is seen as pinnacle of accurate reporting and is meant to convey that Omar voted against HR183.
If he meant by that why "you, the Democrats, including you, Omar" couldn't be convinced to just vote on a resolution that focussed solely on antisemitism, alluding to the way from draft to final version, which in the view of many of the Republicans 'watered down' the initial purpose of that bill, then he may be said to be 'correct' albeit at least hard to follow (whether really towards condemning antisemitism alone, or condemning Omar — an exercise for the reader…)
A 'purified' resolution with just antisemitism in it was then passed by the Senate as Cruz-Kaine Anti-Semitism Resolution.
As FOX-News has put it:
— Senate expressly condemns anti-Semitism, after Democrat-led House 'can't bring itself' to do the same…
In criticising a 'watering down effect', even some Democrats joined the complains.
After debates and delays over the wording of the text, the House on Thursday overwhelmingly passed a resolution that broadly condemned hate.
The resolution – which was crafted after freshman Rep. Ilhan Omar, D-Minnesota, made comments critical of Israel that detractors said played to anti-Semitic tropes – passed 407 to 23.
All Democrats voted in favor of the resolution, including Omar. Republicans were the only to oppose it, with one member voting as present.
The resolution was broadened from its original version, which focused solely on denouncing anti-Semitism, to condemning other forms of bigotry against minorities.
The original effort received pushback from progressive Democrats including New York Rep. Alexandria Ocasio-Cortez, who said it was unfair to single out Omar – although a draft version of the initial resolution obtained by USA TODAY did not mention Omar by name.
The Democrats compromised by drafting a resolution that condemns both anti-Semitism and anti-Muslim discrimination, as well as general "bigotry against minorities." The resolution does not mention Omar by name.
But the resolution did not appear to satisfy some Democrats.
"Why are we unable to singularly condemn anti-Semitism?" asked Rep. Ted Deutch, D-Florida, in a speech Thursday from the House floor.
"This is shameful and unacceptable," tweeted Sen. Rick Scott, R-Florida. "@SpeakerPelosi has shown that Democrats like @Ilhan can make anti-Semitic comments with no real consequences. She continues to sit on the Foreign Affairs Committee and now Dems can’t even muster the courage to officially condemn anti-Semitism."
— William Cummings & Christal Hayes: "House overwhelmingly passes resolution condemning hate after Rep. Ilhan Omar's comments", USA TODAY, Updated 8:23 AM EST Dec 16, 2019.
Or analysed here:
“Why are we doing this?” said Rep. Bonnie Watson Coleman, a New Jersey Democrat, in a closed-door House Democratic Caucus meeting, according to the Washington Post. “We’ve individually and collectively already responded to the fact that we oppose all isms that do not treat people in this country fairly and justly.”
That Omar seems to continue that what some declared to be antisemitic tropes etc seems evident, as seen at the time of these debates over the resolution, thus bringing Cruz's statement halfway on track:
The latest controversy came last week when Omar made a comment about “allegiance” to foreign countries at a bookstore event.
"I want to talk about the political influence in this country that says it is OK for people to push for allegiance to a foreign country," Omar said.
— Rex Santus: "House condemns all sorts of things in resolution originally meant to condemn Ilhan Omar. The resolution decries anti-Semitism, in addition to anti-Muslim discrimination and white supremacy, but it doesn’t mention Omar by name", vice.com Mar 7 2019, 11:34pm.
From a 'progressive' source:
But the lead-up was not easy. The resolution was being drafted up to the moment it was sent to the floor for debate; in fact, debate had to be pushed back in order for resolution authors to update the draft, adding more minority groups that had been left out.
“It’s up to her to explain, but I do not believe she understood the full weight of the words,” Pelosi said. “I feel confident her words were not based on any anti-Semitic attitude, but that she didn’t have a full appreciation for how they landed.”
There’s another reason House leaders were eager to vote on the resolution; they have a big vote coming up on Friday on HR 1, the sweeping anti-corruption bill that was their first bill of the year. This is Democrats’ signature issue, and they do not want the bill’s vote marred by continued infighting.
— Ella Nilsen: "The House easily passed Democrats’ anti-hate resolution, with some Republicans dissenting. Democrats passed the resolution at the end of a fraught week. It came after Rep. Ilhan Omar’s comments last weekend.", vox.com Updated Mar 7, 2019, 5:40pm EST.
— Lindsey McPherson: "House passes anti-hate resolution after days of debate over response to Omar comments. Some Democrats and Republicans wanted a standalone vote to condemn anti-Semitism", The Roll-Call, March 7, 2019 at 5:52pm
He proceeded to criticize Omar for her alleged antisemitism, stating, "And if you want to talk 'literal genocide,' we can discuss the Nazis murdering 6 million Jews in the Holocaust. Why again did you oppose the House resolution against antisemitism?"
Cruz was referring to a US House resolution passed on March 7, 2019 which condemns antisemitism "as hateful expressions of intolerance that are contradictory to the values and aspirations that define the people of the United States."
The resolution additionally condemned "anti-Muslim discrimination and bigotry against minorities as hateful expressions of intolerance."
However, according to the final vote results, Omar voted in favor of the resolution and did not oppose it, as Cruz stated. However, Omar has been scrutinized in the past for presenting antisemitic tropes in numerous social media posts.
— Tamar Beeri: "Ted Cruz slams Ilhan Omar for 'opposition' to antisemitism resolution. The Twitter battle between the opposing politicians began when Cruz retweeted a video posted by ABC News of protesters tearing down a statue of Christopher Columbus.", The Jerusalem Post, June 13, 2020 11:36.
In the end it is the usual: Twitter statements usually aren't accurate, give no room for clear thinking, are to short for decent train of thought or reasoned discussion nor for for any explanations. Almost nobody there ever backs off and admits errors, and this seems much worse for anything party politics. And in this case, of a half-right and half wrong, it leaves a nice projection plane for further polarisation. The triumphant 'Cruz-all-wrong-crowd' aren't showing much desire for analysis but much pleasure in outrage.
