The President has no such authority. There is, in fact, no such thing as "martial law" in the United States.
Martial law on the national level may be declared by Congress or the president. Under Article I, Section 8, Clause 15, of the Constitution, Congress has the power “[t]o provide for calling forth the Militia to execute the Laws of the Union, suppress insurrections and repel Invasions.” Article II, Section 2, Clause 1, of the Constitution declares that “[t]he President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States.” Neither constitutional provision includes a direct reference to martial law. However, the Supreme Court has interpreted both to allow the declaration of martial law by the president or Congress.
When has that stopped him?
There is, in fact, no such thing as "martial law" in the United States.
As we watch the NPS close our national monuments and parks, locking the porta potties, removing the handles from water pumps/fountains.