2016 Postmortem

"If, in foreign invasion or civil war, the courts are actually closed, and it is impossible to administer criminal justice according to law, then, on the theatre of active military operations, where war really prevails, there is a necessity to furnish a substitute for the civil authority, thus overthrown, to preserve the safety of the army and society; and as no power is left but the military, it is allowed to govern by martial rule until the laws can have their free course. As necessity creates the rule, so it limits its duration; for, if this government is continued after the courts are reinstated, it is a gross usurpation of power. Martial rule can never exist where the courts are open, and in the proper and unobstructed exercise of their jurisdiction. It is also confined to the locality of actual war."

So, no, the Constitution would not allow for martial law under current circumstances.