Nations may of course enforce their own laws while foreign nationals are on in their jurisdiction, but once they are outside that jurisdiction, they are at the mercy of the foreigners' home country for extradition. If that country doesn't care to extradite, they're out of luck (barring extraordinary rendition). Delegating the investigation to an international body doesn't change that (not to mention that the US is not a part of and has been actively hostile toward the ICC).

The fact that the ICC counts many nations as members certainly makes it feel like the US should cooperate on moral/ethical grounds, but it has no legal requirement to do so.