Which is fine, inside the US. But when Americans leave the US, they are no longer under US law, theyre under the law of whatever country they enter. Which means, if the country acknowledges the ICC, any American in that country falls under it. Just like a tourist in America has to follow American laws.
See the "Hague Invasion Act"
Not exactly. Tourists are one thing, but US military personnel stationed in foreign countries typically operate under a status of forces agreement which limits the ways in which they are subject to the host country's laws.