By law: US code § 893
See Ridley v Warner 522 F.2d 882 (9th Cir. 1975)
Section 973(b) provides:
"Except as otherwise provided by law, no officer on the active list of the Regular Army, Regular Navy, Regular Air Force, Regular Marine Corps, or Regular Coast Guard may hold a civil office by election or appointment, whether under the United States, a Territory or possession, or a State. The acceptance of such a civil office or the exercise of its functions by such an officer terminates his military appointment."
So election to the presidency, rather than change a civilian into a military man, would do the exact opposite: change a military man into a civilian.
Were the president to be in the active duty military, he could not, under DoD Directive 1344, participate in political campaigns or other activity, or comment on public policy.
