It\'s possible to open the Game Center app from your own app using:
[[UIApplication sharedApplication] openURL:[NSURL URLWithString:@\"gamecenter:\"]];
The example URLs used in Cirrostratus's answer are missing a few bits.
[[UIApplication sharedApplication] openURL:[NSURL URLWithString:@"gamecenter://static.gc.apple.com/me/"]];
[[UIApplication sharedApplication] openURL:[NSURL URLWithString:@"gamecenter://static.gc.apple.com/friends/"]];
[[UIApplication sharedApplication] openURL:[NSURL URLWithString:@"gamecenter://static.gc.apple.com/games/"]];
These URLs seem to work for me on iOS 6 and 7. I'm assuming if you were signed into a sandbox Game Center account that you'd have to use sandbox.gc.apple.com instead. I tried to go to a specific game or friend page via /games/game/
and /friends/player/
but anything I try for ID doesn't seem to work. For the friends URL I go to a blank friend page.
Even if someone did figure it out, because it's undocumented Apple could change at any time in the future so I wouldn't hardcode a URL like this into an app.