Thus the FON OPS we've been conducting. We sail through those seas to demonstrate we don't recognize the sovereignty claims, and the Chinese protest to demonstrate that they haven't given the claims up. It's not just chest-beating - under maritime law, one way to legitimize sovereignty over an area you claim is to show that historically other nations ask your permission to sail through/fly over it, as Quaddafi tried to claim the Gulf of Sidra as a "historic bay" back in the '80's.
I don't think the Chinese will try pushing us deliberately. They have too much to lose and they're not in a position right now to start a war. Where it'll get interesting is when the Chinese start bumping smaller countries, especially the Vietnamese and Filipinos, in an effort to force compliance and rebuff their claims. If the Chinese do nothing, they're implicitly acknowledging the PCA's ruling. If they aggressively push other countries around, they invite an international response they don't particularly want.