***Nothing I am about to post is in any way myself advocating an overthrow of the President or US Government***
The problem with the military taking action is that there is no precedent, or any real guidelines to how it would act. Our Constitution says nothing about holding a coup against our own Government, and how things should be handled.
At the same time, from my perspective at least there doesn't seem to be a strong enough central authority in our Military to do this. It would have to come down from SecDef, JCS, or someone on equivalent level, and I honestly don't see that happening. We aren't going to get some Brigadier or Major General walk up to the White House and say GTFO. They just won't have the authority or backing to do so. And don't even begin to entertain the idea of someone in the enlisted side trying it.
So for it to be done, it would have to be due to the actions of something so appaulling that there could be no question about it. I just don't see anything like that happening during this Presidency.