I would honestly like to see how that would pan out. Obviously this is just a clueless senator pulling things out of his ass for political points, but the worldwide scandal of a US military incursion against an international court of law inside a NATO country would have massive repercussions. It would destabilize US partnerships, diplomacy and it would tarnish US standing even more than it already has been for the last 20 years. Every US opponent country would love for this to happen. This would definitely not go well with EU countries, most of which already said they would stand by the ICC's decision. Arab countries would turn on the US and third world countries probably would too, as the ICC case is massively popular with both those two parties.

I doubt it would spark a war between the US and the EU, but it would certainly sour relationships real damn quick.