Either that, or mutiny. It doesn't have to be a public mutiny, of course - it could be all decided within the Oval Office.
Considering that Israel has nuclear weapons, the possibility of the USA waging a war against Israel is very low, IMO. Obama will not issue an order. If the Turkish ship is destroyed and all on board killed (including the Turkish Prime Minister) NATO could tell Turkey that there is nothing to do until someone invades Turkish land. I doubt that NATO mutual defense agreement will kick in if a NATO country invades some other country and the invasion forces get what is due. Some countries still defend their borders, though that art is largely lost in the USA.
I suspect that US forces would resort to active avoidance.
I assume pilots would not “engage” and would play cat and mouse rather than fight those they consider great and sincere allies. At least, I hope that’s what would occur on both sides.