Do local governments have any control over their local airspace in the US. Could a governor singlehandedly close or restrict the airspace above his state? I know the FAA controls national airspace but could a local government enact local measures? And to what extent could they?
State governors do not have any control over their airspace but may have control over landing sites (read airports). From the FAA's brief on drone regulations:
Congress has provided the FAA with exclusive authority to regulate aviation safety, the efficiency of the navigable airspace, and air traffic control, among other things. State and local governments are not permitted to regulate any type of aircraft operations, such as flight paths or altitudes, or the navigable airspace.
However, these powers are not the same as regulation of aircraft landing sites, which involves local control of land and zoning. Laws traditionally related to state and local police power – including land use, zoning, privacy, and law enforcement operations – generally are not subject to federal regulation.
Yes and no.
Any governance not specifically held by the federal government is held by the state or local governments. The federal government has regulations in place that pertain directly to the National Airspace. State and local governments still retain the rights to govern any part of the airspace not covered in federal regulations. For instance, some state and local governments still allow personal ownership of airspace. Chicago, still has deeded air rights to individuals and corporations.
The administration of Air Traffic Control is a huge, complex, and expensive operation. No single local government can handle the expense. Nor can they handle the complex coordination between each other’s airspace. That, as well as the national security aspect involved, needs to be handled by a centralized government.
Airports are owned by city, state, and local governments. Some are even owned by individuals and corporations. The owners of these airfields can do with them as they please within limits. Meigs Field and Kenneth Copeland Airport are examples of this. That being said, the federal government subsidizes public airports. Since they control the purse strings, they hold a lot of sway over how the airports are run. The local governments can effectively shut down their airspaces by shutting down their airports. But, they may face major consequences from the federal government. This will not stop overflights of the area. An example of this control is the security measures put in place at airports around the country after 9/11. There were city owned GA airports that erected fences around their airfield immediately after at the behest of the federal government.
Think of it like the highway system in the US. It is too big to be left to the devices of local governments.