In a head-to-head fight against the Chinese fleet, the US Navy would get a bloody nose, but the Chinese would be annihilated. Unfortunately, you also have to consider that any fleet engagement would likely be fought with range of Chinese land-based bombers. I think we could take them, but it would get messy. A Guadalcanal-scale messy.
If we go to war with the Chinese, I FIRMLY believe that they would NUKE our carrier forces, then pick off the other ships.
The carriers are a projection of incredible power. I'm sure the loss of a carrier group or two would be worth a couple of cities to the Chi-coms.