Doubtful.

 I like to think that our military leaders are smarter than our political leaders, so they know that no matter how good and how popular the person that they hold the coup for is, they will get blamed for helping them.  Besides, while the president has day-to-day control, the real power lies in congress which has proven itself to be able to frustrate it's radical and reactionaries.

 Political rhetoric aside, none of the current crops of presidential candidates is a complete cretin.  Morally bankrupt of course, but that is to be expected (YES, I am cynical, but look at our choices.  It's a coping mechanism.).  So don't expect any real change in current policies and operations.

 Finally, look at the poll numbers for how much the public respects the various groups.

President -- low double digits

Congress -- SINGLE digit

Military  --  high double digits

Under these circumstances the best way for the military to bring down a president would be to order the troops in the United States to return to barracks, and have the Joint Chief's publically state the their reasons for dissatisfaction and resign en mass in protest.  As long as the troops publically refuse to acknowledge the president and congress (but continue any ongoing operations and civilian rescue/disaster relief) the politicians would be looking at a public relations disaster capable of bringing down ALL parties.

I think it is a safe bet that congress would throw the administration to the wolves (impeachment) to get off the hot seat. 

And by refusing to endanger the public the military gets to keep it's moral authority instead of blowing it in a coup.