The UNILC has produced draft articles specifically on unilateral declarations. The first article of these draft articles states:

Declarations publicly made and manifesting the will to be bound may have the effect of creating legal obligations. When the conditions for this are met, the binding character of such declarations is based on good faith; States concerned may then take them into consideration and rely on them; such States are entitled to require that such obligations be respected

In terms of what constitutes intention to be bound, the bar is actually quite low, as seen through the Nuclear Tests Ban case at the ICJ. In that case the foreign minister of France stated that France was going to conduct its last atmospheric nuclear test in 1973. The statement was this:

'We have now reached a stage in our nuclear technology that makes it possible for us to continue our program by underground testing, and we have taken steps to do so as early as next year'

The official position of its foreign ministry was that France had no legal obligation to no longer conduct tests, and then conducted subsequent tests. Australia took France to the ICJ claiming that Australia had relied on France's declaration in good faith, and France had gone back on its word. The ICJ stated the following (the numbering appears incorrect on my device because I don't know how to remove auto-numbering but it's paras 47-51):

1. Having examined the legal principles involved, the Court will now turn to the particular statements made by the French Government. The Government of Australia has made known to the Court at the oral proceedings its own interpretation of the first such statement (paragraph 27 above). As to subsequent statements, reference may be made to what was said in the Australian Senate by the Attorney-General on 26 September 1974 (paragraph 28 above). In reply to a question concerning reports that France had announced that it had finished atmospheric nuclear testing, he said that the statement of the French Foreign Minister on 25 September (paragraph 39 above) "falls far short of an undertaking that there will be no more atmospheric tests conducted by the French Government at its Pacific Tests Centre" and that France was "still reserving to itself the right to carry out atmospheric nuclear tests" so that "In legal terms, Australia has riothing from the French Government which protects it against any further atmospheric tests".

2. It will be observed that Australia has recognized the possibility of the dispute being resolved by a unilateral declaration, of the kind specified above, on the part of France, and its conclusion that in fact no "commitment" or "firm, explicit and binding undertaking" had been given is based on the view that the assurance is not absolute in its terms,that there is a "distinction between an assertion that tests will go underground and an assurance that no further atmospheric tests will take place", that "the possibility of further atmospheric testing taking place after the commencement of underground tests cannot be excluded" and that thus "the Government of France is still reserving to itself the right to carry out atmospheric nuclear tests". The Court must however form its own view of the meaning and scope intended by the author of a unilateral declaration which may create a legal obligation, and cannot in this respect be bound by the view expressed by another State which is in no way a party to the text.

3. Of the statements by the French Government now before the Court, the most essential are clearly those made by the President of the Republic. There can be no doubt, in view of his functions, that his public communications or statements, oral or written, as Head of State, are in international relations acts of the French State. His statements, and those of members of the French Government acting under his authority, up to the last statement made by the Minister of Defence (of 11 October 1974), constitute a whole. Thus, in whatever form these statements were expressed, they must be held to constitute an engagement of the State, having regard to their intention and to the circumstances in which they were made.

4. The unilateral statements of the French authorities were made outside the Court, publicly and erga omnes, even though the first of them was communicated to the Government of Australia. As was observed above, to have legal effect, there was no need for these statements to be addressed to a particular State, nor was acceptance by any other State required. The general nature and characteristics of these statements are decisive for the evaluation of the legal implications, and it is to the interpretation of the statements that the Court must now proceed. The Court is entitled to presume, at the outset, that these statements were not made in vacuo, but in relation to the tests which constitute the very object of the present proceedings, although France has not appeared in the case

5. In announcing that the 1974 series of atmospheric tests would be the last, the French Government conveyed to the world at large, including the Applicant, its intention effectively to terminate these tests. It was bound to assume that other States might take note of these statements and rely on their being effective. The validity of these statements and their legal consequences must be considered within the general framework of the security of international intercourse, and the confidence and trust which are so essential in the relations among States. It is from the actual substance of these statements, and from the circumstances attending their making, that the legal implications of the unilateral act must be deduced. The objects of these statements are clear and they were addressed to the international community as a whole, and the Court holds that they constitute an undertaking possessing legal effect. The Court considers that the President of the Republic, in deciding upon the effective cessation of atmospheric tests, gave an undertaking to the international community to which his words were addressed. It is true that the French Government has consistently maintained, for example in a Note dated 7 February 1973 from the French Ambassador in Canberra to the Prime Minister and Minister for Foreign Affairs of Australia, that it "has the conviction that its nuclear experiments have not violated any rule of international law", nor did France recognize that it was bound by any rule of international law to terminate its tests, but this does not affect the legal consequences of the statements examined above. The Court finds that the unilateral undertaking resulting from these statements cannot be interpreted as having been made in implicit reliance on an arbitrary power of reconsideration. The Court finds further that the French Government has undertaken an obligation the precise nature and liinits of which must be understood in accordance with the actual terms in which they have been publicly expressed.

A head of state's statements are able to be relied upon in good faith by other states. It is therefore extremely important that heads of state, heads of government and foreign ministers are careful in what they say regarding the policy position of the state. Genocide is a legal concept, and the violation of the prohibition of genocide is egregious. By its nature therefore it is difficult to say that a head of state can make such an accussation but not mean it. I have edited my original comment to include the fact that Biden inclided a caveat and that there may now be a higher bar post-Trump due to his tweets, but it is nonetheless important that he clarify if he did not mean it.

This is often why state departments and foreign ministries scramble to claw back the things said by heads of state. The very works of the head of state can bind the state.

The person you're responding to added a link to a comment that gives some very helpful information on nations' responsibilities to fight genocide under international law.

Re your comment, I would say that Biden calling the situation genocide is definitely not "flippant," and might suggest a formal statement could possibly come. When it comes to international law, the president basically speaks for the nation.

So Biden could certainly make a formal declaration of genocide and state that the United States sees Putin's actions as acts of genocide. When the Administration makes a policy statement, it can be very simple. Yes, it's normally more formal (or more codified) than a verbal statement, but even just a short statement posted online is sufficient for the Administration to essentially direct the executive branch to treat a policy matter as indicated. Also, the United States has no formal policy around recognizing genocide. Congressional resolutions are one way that the legislative branch might recognize genocide, but that wouldn't make it Administration policy. Here is a recent report that discusses how the US has recognized genocide in the past.

Finally, I will add that it is my understanding that if the President recognizes genocide as a matter of Administration policy, the United States would have an erga omnes obligation (as discussed in the linked comment above, I think) to act to end the genocide (e.g., by providing weapons, money, humanitarian aid, etc.) even if it doesn't legally obligate the UN as an international entity to respond. Separate from the President's foreign policy agenda, the UN itself would legally recognize a genocide only through the International Court of Justice. Only then would countries that have ratified the UN Genocide Convention be legally obligated to act as a matter of international treaty rather than an erga omnes obligation.

Hopefully others will correct me if I got anything wrong - this is not my specialty, I just happened to read Samantha Powers's book a couple months ago so have had genocide on my mind.