The wordings of the current
oath of enlistment and oath for commissioned officers are as follows:
For enlisted personnel:"I, _____, 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; and that I will obey the orders of the President of the United States and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice. So help me God." (Title 10, US Code; Act of 5 May 1960 replacing the wording first adopted in 1789, with amendment effective 5 October 1962).
For officers:"I, _____ , having been appointed an officer in the Army of the United States, as indicated above in the grade of _____ do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign or domestic, that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservations or purpose of evasion; and that I will well and faithfully discharge the duties of the office upon which I am about to enter; So help me God." (DA Form 71, 1 August 1959, for officers.)
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Note that each oath is to support and defend the Constitution and
not the President or any other person. While enlisted personnel swear to obey orders from the President and those officers appointed to command them, an officer has no such constraint. It could be argued that if the President issued an immoral order -- say, drop nuclear weapons on Iran in an unprovoked pre-emptive strike -- an officer (most likely the top-ranking General of the affected service) could legitimately disobey the order. Google "UCMJ illegal orders" for different points of view about this issue.