The The United Earth Military (UE Military) was the miltitary arm of the United Earth government. The Military Assault Command Operations was a branch of this agency. Following the formation of the United Federation of Planets, the United Earth Military, along with the Military Assault Command Operations were disbanded.
Balthazar M. Edison's file
In the biographical file of Captain Balthazar M. Edison, his service in the United Earth Military was noted. This file was accessed by Captain James T. Kirk in 2263 of the alternate reality. (TOS movie: Star Trek Beyond)
Sometime after the Burn, Earth withdrew from the Federation, and Starfleet's presence on Earth began to fade. This led to the founding of the United Earth Defense Force. (DIS episode: "People of Earth")