You need to be more specific. You need to tell me what this hypothetical constitutional violation is. You and I live in a country that has been violating the Constitution almost every day for most of our lives. Hell, it can be argued (not necessarily by me) that Dubya probably did it with this war.
The last CW was over rebellion. An attempt to Balkinize North America over the right to own human beings for profit. By people trying to break up the United States.
In that case, doing anything but fighting for the United States would have been a violation of my oath.
But lets not get off on a tangent. I don't want this to devolve into a "the CW was not about slavery".
Sticking to the larger point, you forget FRiend that the oath we are discussing also says "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."
That POTUS would be Obama and God would be listening.
If the order said to round up all undesirables, and it listed conservatives and business owners as undesirables, would you follow it?
Judging by your tagline, this might not apply, but here goes, anyway.
Hypothetically speaking, let’s say the president has issued an executive order that all firearms in private (non military/LEO/aproved agency hands) be confiscated. Some comply, many do not. A particular group of holdouts not only does not comply, but is vocal about it.
The president orders military forces to confiscate the firearms, by force if needed, authorizes lethal force, house to house searches, etc. in order to accomplish the task.
If you were in the unit so assigned, would you comply with the order. If not, would you oppose it? How far would you be willing to go to defend the Second Amendment of the Constitution?
Next question. Suppose it was incontravertably revealed that (just for fun) the president was not Constitutionally qualified to be POTUS: that in fact, he was not a natural born citizen. As your oath is first to the Constitution, and his occupation of the office and usurpation of its powers would be an egregious violation of the Constitution, how would you react to that?
Remember that takers of said oath are also legally required to refuse to obey illegal orders.
Perhaps you'll recall the "I was only following orders." defense has generally not been a winning strategy.