I'm not sure what court would have had applicable jurisdiction. The International Court of Justice (IJC) would really be the only existing international court at the time. And it has to refer all cases to the UN Security Council for enforcement. And the US has a veto there.

And in general the US hasn't accepted the authority of external criminal courts (like the International Criminal Court) and hasn't ratified their establishing charters. And I can't think of another one that was in existence at the time anyway. The ICC was established in 2002.

Functionally there was no one to press charges other than the IJC. And as they would have basically had to ask the US for permission to go forward, that was clearly a non-starter. So outside of increased hostilities from other countries there was nothing anyone outside the US could do practically to Kennedy or his cabinet members.