Also, because they are tired of the diplomatic cost and expense of working with other countries to set up ad hoc tribunals for particular conflicts and want to get the job done once and properly. (That's actually why the US was one of the leaders of the effort that produced the ICC, even though it did a U-turn against it at the last minute.)
To answer the GP more directly: Treaties the US enters into are part of US law.
Same reason individuals tend to want to live in a society and the rules that come with it.
In this case, there’s a straightforward benefit to it in that it could be used to prosecute crimes against the US and US citizens, and soft benefits e.g. of the US being seen a a paragon of lawfulness and trust. There’s likely more, these are just what I could think of immediately.
Since the leader (elected or not) generally controls the executive apparatus (that’s what makes them the “leader”), its unlikely that this would actually occur, irrespective of the nominal obligation. It is more likely that they would be turned over after being removed as leader (a process which an ICC warrant may or may not accelerate, depending on the response of other institutions, the populace, etc.)
At least, that is what has generally happened with the top leaders that have ended up at the ICC (or its predecessor ad hoc tribunals, back to Slobodan Milosevic, the first sitting head of state to be charged with war crimes by an international tribunal—the ICTY—who wasn't turned over by Yugoslav authorities until after he resigned.)
> because Europeans disagreed with a military decision
The majority of State Parties to the Rome Statute are non-European, as are a majority of the current judges of the court, as is the current acting chief prosecutor, so it is really weird to describe ICC involvement this way.
Would GWB have acted differently, had he been forced to consider the legality of his actions under a broader framework? Would we have preferred for presidents to have even broader immunity from prosecution?
Article VI, Clause 2: This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any thing in the Constitution or Laws of any State to the Contrary notwithstanding.
On a national level I agree not to steal, and in return nobody else is allowed to steal from me. On the ICC level my country agrees not to genocide anyone, and in return others aren't allowed to genocide either
Creating a global enforcement arm is an obvious non-starter: Which laws shall be enacted and enforced against whom, and how are fundamentally political questions. The answers to those questions necessarily change once you cross into another polity. States can coordinate and agree mutually to enforce a law... but if Poland says that despite their commitments to the contrary they simply aren't gonna extradite that suspected Nord Stream saboteur: Tough nuggets. Such situations expose international "laws" as mere window dressing on powers that emanate from and remain held by individual states.
There is no global enforcement arm and plenty of evidence that if someone starts genociding you, nobody else is gonna do anything about it. You're on your own. Such "laws" bind but do not protect.
Especially in the context of the US as world hegemon: The US fully expects to shoulder the entire burden of deterring or defeating threats to the homeland. Joining the ICC would bring precisely zero extra protection from genocide. Indeed, such a commitment may limit the US's freedom of action, discredit US deterrence, and actually make a devastating war more likely.