As formulated: yes. But it's more complicated. Homeopaths did not invent this principle. They interpret it in a peculiar and very far reaching way.
First, we should note that 'Hippokrates' is somewhat elusive as a person or 'writer', since the writings we categorize in the corpus hippocraticum are a large collection of texts spanning a timeframe exceeding any however clever and long-lived individual that also suffer a bit for this kind of authenticity:
Of the texts in the corpus, none is proven to be by Hippocrates himself…
The word Hippokrates is therefore a shorthand we use to ascribe one person to a medical text corpus. The question therefore becomes:
Is there a statement similar to 'like cures like' in that text collection?
The answer to that is a slightly superficial 'yes', (usually claimed to be "Morbo sacro 18" and "De locis in homine 42") greatly explained and summarized here:
Numerous medical traditions—including the Hippocratic writings— propose that “like cures like,” the homeopathic principle. “Homeo-” is from Greek homoios, which means “similar,” and “-pathic” is from Greek pathos, meaning “suffering, disease”; thus, “homeopathic” means “similar to the disease.” Briefly, the homeopathic principle can be construed in a more general, noncontroversial manner or in a much more specific (and controverted) fashion.
In its most general form, the homeopathic principle would simply have one treat a patient suffering from a disease that produces, for example, heat as a symptom (say a fever) with something hot (rather than something cold). The homeopathic principle in its more specific (and disputed) formulation derives from the occasionally noted phenomenon that substances that in healthy people induce symptoms like those caused by an illness at times cure sick people of that very illness. With this phenomenon in mind, the German physician Samuel Hahnemann coins the term Homöopathie (from which we get “homeopathy”) in 1807. Like cures like. That which produces in healthy people symptoms similar to those of a specific sickness sometimes restores health in those who suffer from that ailment. Hence, the homeopath proposes to treat the sick with that therapy. Lest I be misunderstood as an advocate, I do not propose or endorse an overarching homeopathic approach to medical practice—as, for example, Hahnemann does. Rather, I note that an ancient practice-based observation (homeopathy) has deeper implications, […]
The French medical historian Jacques Jouanna brings to our attention a text from Hippocrates: “Another principle is the following: a disease arises because of similars, and, by being treated with similars, patients recover from such diseases” (Hippocrates 1995, 83). Jouanna writes, “therapy by similars was already sufficiently known in the fifth century b.c. to have found expression in the theater with Sophocles: physicians: ‘evacuate bitter bile with a bitter clyster’” (Jouanna 1999, 473–4). As Jouanna reminds us, while the Hippocratic corpus recognizes the occasional efficacy of homeopathy, “Hippocratism is … founded on allopathy, or treatment by agents producing effects contrary to those of the disease (Jouanna 1999, 343).” As will be noted at greater length, the terms “allopathic” and “homeopathic” date to the 1800s.
— T.A. Cavanaugh: "Hippocrates’ Oath and Asclepius’ Snake. The Birth of the Medical Profession", Oxford University Press: Oxford: New York, 2018, p15 & p158.
One precise example — well known to this day, perhaps — is found in Epidemics 2
"If, following intoxication, there is a headache, drink a cotyle (= 0,27 litres) of undiluted wine"
— Hippocrates, Epidemics 2, 5th sect., ch.30, 5.138, 9f.L. Quoted after:
— Jacques Jouanna & Neil Allies (transl) & Philip van der Eijk (ed): "From Hippocrates to Galen. Selected Papers", Studies in Ancient Medicine 40, Brill: Leiden, Boston, 2012. p184 (Loeb version L477 Epidemics Book2 on archive.org)
Sidenote: the full Latin adage similia similibus curentur appears only once in this complete form in 27000 original pages of Hahnemann's work. The shorthand similia similibus or variations are then numerous. Its meaning is not "like cures/heals like", but rather "treat like with like" (cf medicus curat, natura sanat, the doctore cares, nature heals.).
In the Hippocratic writing "On the Places of Man" (Peri tópon tón kát' ánthropon, De locis in homine), three different treatment principles (the Simile principle, the Contrarium principle and the principle of "sometimes this, sometimes that") are examined for their generalisability. In contrast to Hahnemann, who quoted this book — albeit selectively — in the "Heilkunde der Erfahrung" and in the introduction to the "Organon", the Hippocratic author comes to the conclusion that there can be no fixed rule at all in medicine that always applies without exception. Rather, everything depends on the individual case and the respective context, even on the unique moment, in Greek "kairös". Principles and theories therefore always have a purely instrumental character, and their artistic selection and application to the individual patient is up to the judgement of the healing artist.
— Josef M. Schmidt: "Samuel Hahnemann und das Ähnlichkeitsprinzip", in: Robert Jütte (ed): "Gesellschaft und Geschichte. Jahrbuch des Instituts für Geschichte der Medizin der Robert Bosch Stiftung, Vol
29, Report Year 2010, Franz Steiner: Stuttgart, 2011. p176 own translation from German.
A detailed discussion about ancient words, their original meaning, similiarities to Hahnemann's theory and his interpretation is found in:
It would be wrong, on the basis of the attention paid by the empiricists to 'healing by means of the same', to see in them something like homoeopaths of antiquity. For although both show a certain agreement in their appreciation of experience and phainomena, as well as in their rejection of aitiological statements regarding the adela, the ancient empiricists differ substantially from modern homoeopathy in their assessment of the 'same': For them similia similibus is neither a regulative nor a heuristic principle of therapeutics; it possesses only an eristic, anti-dogmatic meaning.
In late antiquity, already on the threshold of the Middle Ages, one finds Isidore of Seville's Origines, which represent the sum total of what was still known of ancient science at that time. In it the following sentence is found in an outline of medicina: omnis autem curatio aut ex contrariis aut ex similibus adhibetur.
There, the empiricist's antithesis of De locis in homine, which led to scepticism, became a new dogma by addition, as it were, which unites both modi medendi and as such claims general validity (omnis autem curatio). The words are almost the same as in the Hippocratic writing, which is a thousand years older, but the underlying meaning has turned into its opposite.
— Carl Werner Müller: "Die Heilung „durch das Gleiche“ in den hippokratischen Schriften De morbo sacro und De locis in homine", Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften, Bd. 49, H. 3, September 1965, pp225–249 (jstor, own translation from German).
The original Hahnemann quotes is for "De locis in homine 42":
So sagt der Verfasser des unter den Hippokratischen befindlichen Buchs [Über die Stellen des Menschen] die merkwürdigen Worte: [Durch das Ähnliche entsteht Krankheit und durch Hinzufügung des Ähnlichen werden aus Kranken Gesunde … Durch das Erbrechen hört das Erbrechen auf.].
— Joseph m Schmidt (ed): "Samuel Hahnemann: Organon der Heilkunst als Praxisausgabe", Urban & Fischer: München, 2006. (ancient Greek omitted)
These words (which include ancient Greek for the text in Brackets) translate as: "Thus the author of the book [On the Places of Man], which is among the Hippocratic, says the strange words: [By the similarity disease arises, and by the addition of the similar the sick become healthy … By vomiting vomiting ceases]."
Hahnemann uses an old version/edition (Basil. Froben. 1538. p72.) of the transmitted text and most early printed versions contain numerous small errors of the Greek, but without changing much of the underlying meaning.
This original Greek quote used is found in a correct version in:
— Émile Littré: Oeuvres complètes d’Hippocrate, traduction nouvelle avec le texte grec en ragard, collationné sur les manuscrits et toutes les éditions. Baillière: Paris, 1849, Vol. 6, p. 336, l. 3 and p. 334, l 11 f. Greek text with French translation ("Mode de production de la douleur. Guérison par les contraires; guérison par les semblables.") on: archive.org)
An English version of Places in Man, 42, reads as:
42 Pain arises both from cold and from heat, and both from excessively great amounts and from too little. In persons that are cooled by nature out of their body towards the skin, pain arises from excessive heating, in those by nature hot, from cold, in those by nature dry, when they are moistened, and in diose by nature moist, when they are dried. For in each thing that is altered with respect to its nature, and destroyed, pains arise. Pains are cured by opposites, and there is a specific thing for each disease: in persons by nature hot, and who are ill because of cooling, it is what heats, and so on according to this principle.
Another principle is the following: a disease arises because of similars, and, by being treated with similars, patients recover from such diseases. For example, the same thing produces strangury when it is not present, and stops it when it is present; cough, in the same way as strangury, is engendered and is halted by the same things. Another principle is the following: fever that has arisen due to phlegmasia sometimes arises and is stopped bv the same things, and sometimes is stopped by things opposite to those from which it arose. For if someone washes this patient with hot water and gives copious drinks, the patient recovers as a result of this swelling; when things that promote swelling are administered, the fever that is present becomes well; also if someone decides to make this patient drink a downward evacuant and an emetic. Thus, in the same way it is stopped by the things that produce it and produced by those that stop it.
That is to say, if to a patient that is vomiting someone gives much water to drink, the patient will, by vomiting, be washed clean by what he vomits; thus, through vomiting, the vomiting stops; and in this way through its stopping—because it will make the patient evacuate downwards that which, when present in the body, provokes vomiting—the patient recovers in two contrary ways. And if this were so in all cases, the principle would be established, that sometimes conditions can be treated by things opposite to those from which they arose, and sometimes by things like to those from which they arose.
— Paul Potter: "Hippocrates VIII", LCL 482, Harvard University Press: Cambridge, Massachusetts, London, England, 1995. p85–87 (archive.org)