Have the same issue on iPad Air 2. The upgrade to iOS9 went fine on iPhone 6 but on the iPad Air, the Game Center shows blank and selecting it in Settings just freeze the Settings app.
Had five contacts with Apple over the last three days. Three times via chat, then two consecutive senior support engineers over the phone.
Issue is now emerging at Apple and they start to take it seriously. After trying the usual (hard reboot, reset all settings, restore to iOS8 and re-upgrade, restore to blank iOS9 + load backup), it all failed and so got escalated to the senior guys who said the issue was incompatibility between iOS9 and "something" in my backup data... Indeed it makes sense because GameCenter loads fine under blank iOS9 and only starts failing when I put my backup back on (whether from iTunes of iOS).
Then they said:
"Problem is known and being examined. You have two options: #1 is to restore to blank iOS9 and DO NOT LOAD any backup for a few days until we work on the iCloud backup data and let you know when you can restore it (probably at the same time as a fix is issued) or #2 is to restore to your latest iOS8 full backup and wait for the fix to be issued before upgrading again to iOS9.
So there you have it.
A fix is coming and we have to decide whether we want to go back to iOS8 with our data, or stay on iOS9 but without our data, both until a fix is coming...