This is an interesting read that references a lot of court rulings on the matter of martial law in the u.s.

The long story short of it:

"The Constitution gives Congress authority to regulate the domestic deployment of the military, and Congress has enacted comprehensive legislation in that area. Because that legislation does not include authorization for the president to impose martial law, the president has no power to do so. Even if Congress were to provide authorization, the Supreme Court has not conclusively decided that the federal government is constitutionally empowered to declare martial law. But it is clear that such a power, if it existed, would have to be exercised within the bounds of the Constitution, and the military’s actions would be subject to judicial review.

The state martial law power is more clearly established, but there are significant limits. States may declare martial law whenever it is authorized by state law, and federal courts are likely to defer to a state governor’s decision that doing so was necessary. However, the Constitution and valid federal laws will still constrain states’ conduct under the declaration, and judicial review will be available in federal court.

Moreover, even without the power to declare martial law, the president still has extensive authority to deploy the military domestically to perform law enforcement functions. The Insurrection Act — and possibly Title 32 as well — leave it almost entirely up to the president to decide when and where to use U.S. armed forces at home against Americans. To some observers, a deployment of troops under the Insurrection Act might look and feel very much like martial law. Given the degree of confusion over the term, some within the media or the government itself might even call it martial law. Although that label would be inaccurate and the military’s authority would be substantially less extensive than under martial law, the fact remains that any use of U.S. armed forces as a domestic police force represents a departure from American tradition and carries inherent risks. Thus, as with martial law, Congress urgently needs to pass legislation clarifying and limiting the president’s powers under the Insurrection Act and Title 32"

Martial Law in the United States: Its Meaning, Its History, and Why the President Can’t Declare It