Game Center integration requires that you break up your main loop in order to yield control back to the system. In other words, instead of running an endless main loop, you run each frame in a callback function, using:
int SDL_iPhoneSetAnimationCallback(SDL_Window * window, int interval, void (*callback)(void*), void *callbackParam);
This will set up the given function to be called back on the animation callback, and then you have to return from main() to let the Cocoa event loop run.
... do frame logic and rendering
int main(int argc, char *argv)
... initialize game ...
// Initialize the Game Center for scoring and matchmaking
// Set up the game to run in the window animation callback on iOS
// so that Game Center and so forth works correctly.