So, folks on twitter last night started asking about a coup as the solution to the Trump Problem. Civ-mil twitter plus Americanists (those who study American politics) rose up instantly and said: Hell no! It is rare to see such an idea that combines stupidity, ignorance, and norm-violation ... unless it comes out of the Trump White House. Because I have a CDSN event this afternoon (see more this evening or tomorrow), I will be relatively brief in my answer.* To preview: the US military can't, it won't, and if it did, it wouldn't fix things, and it would lead to more coups.
There are not battalions of troops near the White House ready to take over. The US military is not deployed in the national capital, so who is going to coup? Who is this military you want to see power? Folks mentioned the Chairman, which shows how ignorant they are. The Chairman of the Joint Staff does not have operational control or command over anyone except the Joint Staff, which are desk folks in the Pentagon, not units of armed people capable of seizing the White House, Congress, TV/Radio stations, and the other spots that are targeted in coups (see either Naunihal Singh's book or the game of Junta). How would the coup work? Who would command? Who would be deployed? The key, as Singh would argue, is to create a sense of inevitability of victory on the part of the coup plotters. Since this has never been done in the US, how does that happen?
There is no history of the military doing anything like this. It runs against all over their training, instincts, doctrine, habits, and identity. While the US military would prefer to be autonomous, they drink and breathe civilian control of the military. It is about as thinkable as cannibalism.
Plus, well, the senior military knows too well that it is divided on the merits of this administration. So, there would be the risk of elements of the military firing on the other elements, so that is another motivation for not doing it. The US military loves being the most popular, respected institution in the US. That would end instantly with it getting seriously involved in who governs. Remember the election crisis of 2000? Did anyone ask the military what it thought? No. It all went to the courts.
Third, won't fix things. Pretty much every armed force that takes power will claim that it is fighting corruption and incompetence, but, within a short period of time, everyone learns that militaries are quickly corrupted and lousy at governing. In short, militaries suck at governing, and governing sucks for the military. Oh, wait, they promise to turn it over to the politicians ASAP, maybe after an election. That is what they promise, but they often find that either the process is bad, or the outcome is bad, so they stick around. (No, I am not subtweeting Egypt. Ok, just a bit, but Egypt is hardly alone in this).
Finally, coups lead to .... coups. While there are many factors associated with coups (there is an abundant quantitative literature), one of the most strongest correlates is if one had a coup before. It might be partly because the same structural conditions that caused a coup at time t also caused a coup at time t+1, but it is also that the first coup changes the norms, making seizing power imaginable and not entirely taboo, it teaches the military how to coup, and so on.
So, no. While the US is broken, and its institutions are breaking down, let's hope that the 2020 election solves the Trump Problem. We can't count on impeachment or the 25th Amendment since the GOP is craven, but we also can't count on the US armed forces to solve our political problems. It wouldn't work, and it would do irrevocable damage to the military and to the country.
* I am not citing anything due to the aforementioned CDSN event imposing a time constraint. But if you have any suggestions for stuff I should cite, let me know.