Clearly, anyone plotting a coup against the duly elected Pesident would not be fit to hold office, and should be prosecuted accordingly.
At the end of the movie, the Joint Chiefs were forced to resign by the president... the official reason given to the public was their ongoing opposition to the treaty (I think at least General Scott was seen publicly criticizing the president's decision to sign the treaty earlier in the movie).
No charges were pressed for the attempted coup, because Pres. Lyman thought that it would hurt the country.