The way we interact with the Internet would have to change though due the extra delays involved in communication between planets.
A lunar base would have a delay of about 2.5 seconds, so could probably just connect direct to the Internet on Earth via a few gateways located on the Earth (possible just a simple SOCKS proxies with a high value on the TCP Timeouts on it's lunar side). It would be slow, so you'd want to include quite a bit of caching for data on both sides that doesn't change.
Once you start talking about other planets though your delays would start to become unusable for interactive operations (between 5 and 20 minutes for Mars), so at that point you'd have to start thinking about using more asynchronous methods of communication (e.g. send an email rather than use a chat session).
I have to admit that part of me loves the idea of using a variation of FidoNet for interplanetary communication :)