We actually were a signatory (and one of the prime movers for regularizing the ad hoc processes which we had also been a prime mover for creating in Yugoslavia and Rwanda) of the Rome Statute under Clinton, though the signature was effectively withdrawn under Bush.
Not a great look.
I looked into it more and while 7 voted against, 21 abstained so votes don't tell the whole picture. Between China, Indonesia, India and The US you're talking a major chunk of the world population.
Are there potential circumstances where the US Constitution might conflict with obligations under the Rome Statute? Maybe; that certainly fairly regularly happens in practice with the Vienna Convention on Consular Relations, but the US has signed and ratified that treaty.
They obviously do not. If they did, you could basically abolish the Constitution with one with only the presidency, a bare majority of the Senate, and the cooperation of a foreign power. An actual Constitutional amendment must meet a much, much higher bar.
Congress does get extra power when a law is passed as a treaty. Issues that are state business under the 10th Amendment become federal business.
But it’s nothing like a constitutional amendment.
