The oath is to protect and defend the Constitution and to OBEY the LAWFUL orders of the PRESIDENT of these United States. While the first part of the oath is to protect and defend the Constitution in principle; the second part of the oath is to OBEY lawful orders in practice.
While the oath is not sworn TO either the Constitution or the Presidency (it is sworn to God), it obligates one to obey the lawful orders of the President.
appointed a (insert rank) in the U.S. Army under the conditions indicated in this document, do accept such appointment and do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic, that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter, so help me God.
U.S. Military
This is the officers oath. The enlisted oath references the POTUS