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He could argue it, but he'd be clearly wrong. Trivially because there are other officers in the chain of command who can (and do) give orders to Seal Team 6. More importantly, because the fact that the POTUS gives the order doesn't make it an official act exercising executive authority. If POTUS orders Seal Team 6 to pick up Tiffany's dry cleaning or man the phone banks at his re-election campaign, that's not part of the executive function under the Constitution (nor is it an authorized power). So it wouldn't be an official act, and neither would a political assassination of a U.S. citizen on U.S. soil.
And this is my understanding too, thought it took a bit to find some clarity in my mind because there's such emotion behind this.
But I remember Jefferson writing that there are times when circumstances warrant quick action to preserve life, liberty, and property even if not withing Presidential powers at the moment. That a President must make those decision, but manning phone banks isn't one of those times.
But Tiffany's dry cleaning? Hmmm.... :)