No, but they regulate the part where a chunk of white-hot metal and composite plummets into an area of the sky they're responsible for. That extends well beyond U.S. territorial airspace--by international agreements, much international airspace has been assigned to various nations to control. If SpaceX were to conduct their entire reentry out in the middle of the pacific in completely uncontrolled areas then they probably wouldn't need a license, but recovering the capsule would be a whole lot harder.
Do they also require a license to exit the planet through said airspace? Is it possible to receive one but not the other? i.e. If by some clerical error SpaceX received their "Exit License" but then forgot their "Re-Entry License", would the capsule then be stuck in the outer atmosphere?
Pedantic, I know, but still oddly intriguing to me. The fact that such a legacy organization like the FAA has to stretch to fit into this business space is interesting to me.
They do need a license to launch, and such licenses are very common: commercial satellite launches happen all of the time. The FAA needs to be involved because those rockets pass through controlled airspace on the way up and the FAA needs to take measures to keep aircraft away from them. This is news because, for the first time, a commercial entity is going to attempt to bring something back from orbit instead of just sending it up and leaving it there.
Reading between the lines, SpaceShipOne apparently didn't need a re-entry license, probably because they never achieved orbit and so their return didn't meet some technical criteria to be considered a "re-entry" by the FAA. They probably had some sort of special license from the FAA, though.
Just try launching/landing a space vehicle without informing air controllers in any country with a functioning government. I'm pretty sure black helicopters would be near you in minutes, and stern-faced guys in battle gear would start yelling questions.