They wouldn't have to - the VP would become the President and therefore the CIC. He would instruct to military to remove the former president. If the VP does not oblige, he can be impeached as well and the Speaker of the House would become POTUS.

At that point, being the ones with all the guns, it would fall to the military to decide who they are loyal to: the Constitution, ex-POTUS, or themselves, and they will act accordingly.