From what I recall of the UCMJ, a court martial could do exactly that. They could also find the officer not guilty in the (likely) event that the White House failed to produce such a document upon request. Obama cannot afford to have a military officer openly questioning his authority, particularly if that officer then gets away with it at a trial.
Article 90. Subsection 2.
“Any person subject to this chapter who—
2) willfully disobeys a lawful command of his superior commissioned officer; shall be punished, if the offense is committed in time of war, by death or such other punishment as a court-martial may direct, and if the offense is committed at any other time, by such punishment, other than death, as a court-martial may direct.”
(2) Disobeying superior commissioned officer.
(a) Lawfulness of the order.
(i) Inference of lawfulness. An order requiring the performance of a military duty or act may be inferred to be lawful and it is disobeyed at the peril of the subordinate. This inference does not apply to a patently illegal order, such as one that directs the commission of a crime.
(ii) Authority of issuing officer. The commissioned officer issuing the order must have authority to give such an order. Authorization may be based on law, regulation, or custom of the service.
Therein lies the rub. His orders do NOT come from the President, they come from his commanding officer.
There is no good side to this, and a Courts-Martial is inevitable.
He is brave and a patriot, but he is also terribly misinformed of his standing in this issue.
You must be a civie? It will never reached the Whitehouse. The soldier will receive a court martial and given a dishonorable discharge. He will be ruined for life from working in any capacity that requires a criminal background check. He will have a permanent entry in the FBI NCIC database. At this point his life will be over - was it worth it? Think about it...
http://www.fas.org/irp/agency/doj/fbi/is/ncic.htm