No, Bush couldn’t. It’s against the law.
"State Defense Forces (SDF) (also known as State Guards, State Military Reserves, or State Militias) in the United States are military units that operate under the sole authority of a state government; they are not regulated by the National Guard Bureau nor are they part of the Army National Guard of the United States.
State Defense Forces are authorized by state and federal law and are under the command of the governor of each state. State Defense Forces are distinct from their state's National Guard in that they cannot become federal entities.
The President of the United States is also its most powerful military leader known by the term of Commander in Chief, and is authorized by the Militia Act of 1903 to federalize his individual States National Guard Forces putting them under his command, something he does not have the power to do with State Defense Forces.