Not necessarily, he may be operating on his own behalf, but what makes him unlawful is the lack of gov't sanction and being recognizable in the field.
If encountered in a engaged in hostilities they may be killed and if captured tried for war crimes.
I refreshed my memory of the law by reading the Supreme Court case in Hamdi, and the concurring opinions. It was a 4 judge plurality, so how it would come out with a new court remains to be seen. But the way it was written, Hamdi’s trial by military was tied to the existence of a Congressional resolution of military action against radical islamic terrorists. Whether military tribunals could be extended to actions by US citizens on behalf of general bad actors who are not involved on the other side of an actual war remains to be seen. That interpretation would stretch existing case law. The discussion between Kavanaugh and Lindsey Graham was entirely in the context of combat authorized by Congress—WW2 and the War on Terror declared after 9/11.