I am no expert and only repeating some fact sheet a news outlet posted.
I think it would have to be indirect via the UNSC as the UNSC can refer motions re war crimes to the ICC. The US will likely argue that because Israel is outside jurisdiction, no arrest warrants can be issued for the conflict. As in Hamas arrest warrants can’t be done, but neither can Israeli arrest warrants. That none of the conflict is anything the ICC can rule on. They would expect to get broader support that way.
They will argue Hamas leaders, once caught, should stand trial in Israel for their war crimes under their domestic terrorism laws (which carry the death penalty.) A few Arab nations might be ok with this, as they can whisk away Hamas leaders with no extradition and protect them too.
If the US loses this argument (high chance rn) and can’t get the UNSC on side to either agree or abstain, they will then whisk Nethanyu and Gallant away into exile in the US (US isn’t under ICC jurisdiction and there is no extradition agreement) and then say, oh look new leadership in Israel, let’s look to the future.
The US has a lot of soft power it can deploy to make states vote the way it wants them too. Economic sanctions, withholding of foreign aid, pledges to intervene or stay out of other conflicts. I wouldn’t put it past the US telling Putin we will let you have Ukraine if you vote for the ICC to drop this..