They actually use the NRO's Quasar relay satellites for this. They don't connect to the "internet," but rather to NRO mission ground stations, but they need that single point of ingress to Earth anyway because the hardware decryption modules, algorithms, and key-loading mechanisms only exist in military comms equipment, not IP routers. From there, provided the data itself is unclassified or can be downgraded, it can get to the Internet.
It's arguably an interesting question whether the government would consider using commercial relay satellites instead of just the Quasar constellation, though. The data stream doesn't need to be decrypted on the satellite, just forwarded. Obviously, you can't prevent radio from being intercepted, so throwing in a hop you don't own doesn't actually add any risk. You're totally reliant on the strength of your encryption either way.
Actually, it wouldn't be. Starlink orbits at about the same altitude, but the satellites have their radio antenna pointed downwards to Earth, so they can't connect with each other.
Last I heard laser links were in testing, and was currently only being used for communicating in the same orbit (a single, linear string of satellites all orbiting in the same plane and at the same altitude)
