I'd recommend signing out of Game Center, force closing any open apps, then restarting your iPad.
You can sign out of Game Center at the following location:
- iOS: Tap Settings > Game Center.
- OS X: Click your Game Center account on the Me pane, and then View Account.
I have this same issue after updating to iOS 7.0.6. Game Center is a complete white screen. Cannot sign out of Game Center from "Settings > Game Center" either. All that shows up in Game Center settings is my Apple ID and the spinning wheel. Tapping my Apple ID here (to sign out) has no effect. I even tried resetting my Apple ID password to trick it into failing, but no dice.
Restarting/resetting my iPad does NOT resolve the problem. The only temporary solution is to reset all settings: Settings > General > Reset > Reset All Settings. It's a temporary solution as the problem returns in a day or two. Very frustrating, as reconfiguring all settings with that frequency, just to access Game Center, wastes even more time than playing the games I can't without Game Center.
I don't know if there is an application compatibility issue, they all seem to work after resetting all settings... It all started happening after upgrading to 7.0.6... Looks like a lot of people are having this issue, though for some it started after upgrading to earlier versions of iOS 7.