The oath is as follows:
"I, (state your name), 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{{, the governor of the state of _______ (for National Guard enlistees)}} 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).
http://en.wikipedia.org/wiki/Oath_of_enlistment
Watada swore to "bear true faith and allegiance" to the Constitution, which requires him by extension (he says) to not violate the UN Charter, since the US Constitution gives equal weight to treaties as it does to itself.
I don't know if he would accept the US Supreme Court as final arbiter of Constitutional conflicts.