Is it possible for a member of the US military to refuse to participate in an armed conflict that is not being wage in defense of the US Constitution?

Article 1, Section 8, Clause 11 (the Wars Power clause) grants congress the right to declare war and Article 2, Section 2, Clause 1 gives the US President the title of commander in chief.

The legitimacy of military power in the US derives solely from the US Constitution. If war is waged in a way that does not support the Constitution, I would imagine it should not only be every soldier's right, but duty to disobey an order.

The Armed Forces Oath even suggests that should be the case:

"I, (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 and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice. So help me God."

The Oath states that they are to defend the Constitution, not Congress, not the President, not Wall Street, etc.

When soldiers take the Oath, are they also required to learn the Constitution in detail? If not, how are they to know whether or not their actions are in defense of it?

I do not see how this is relevant. Perhaps you could explain that to me.

It is OT from the OP and probably a better to ask this on Quora, but your comment led me to ask this question

Fair enough. Afraid I don't know the answer.

I guess we're pretending the part about obeying orders isn't in there, then?

Weird. I had edited it to reference that, but my edit was saved. Doh!