If one does not read the Constitution one might come to that conclusion based on the President being CinC. If one reads the Constitution, specifically Article I, Section 8, then it's clear that that conclusion is wrong. Congress is the source and the authority for military regulations and military law.
Winthrop addresses this in distinguishing between the constitutional functions of Congress and the President.
"The word "regulation" or "regulations," (as also the allied term "rules,") is employed sometimes in the Constitution as descriptive of statute law, and this use has proved confusing to the student. A similar designation occurs in certain, especially of the earlier public Acts, though it is not frequent. As a general practice, Congress, in framing a public law in which provision is made for an elaborated system, a measure of policy, or other extended subject or project, of which the execution involves minor details of performance, disposes of such details in one of three forms. It either goes on itself to prescribe rules, general or specific, for such performance; or it authorizes some public oficer to make proper rules for the purpose; or it is entirely silent on the subject, prescribing no regulations itself and devolving no authority, in terms, upon any official."
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"The Constitution devolves it upon the executive department to "take care that the laws be faithfully executed." In a case, therefore, of a law of which the execution requires to be specifically methodized, it is the duty of that department, and it is authorized, in the absence of any express authority for the purpose, to prescribe the rules or directions necessary and proper to effectuate the object of the statute; care of course being taken that the regulations shall not partake of the nature of legislation. This inherent authority of the Executive -the President, or head of a Department acting for and representing him - has been repeatedly noticed and affirmed by the authorities."
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"The authority for army regulations proper is to be sought primarily in the distinctive functions of the president as Commander-in-chief and as Executive."