If you're talking about Northern Philippines islands in South China sea, than it would most likely be China that wins. US forces attempting to seize back island would get overwhelmed by Chinese land, sea and air forces defending the island. Any attempt by USA to build up necessary infrastructure and forces on Philippines would take time. Time that China wouldn't allow and would invade Philippines mainland in pre-emptive operation. Most likely scenario would be few USA bombs than a Cold war style stalemate with proxy wars.
If you're talking about Islands in Philippine sea down south: USA would win as China can't project power that far out, compared to USA. It would have to invade Philippines mainland, which would give USA time to digg in and bring in defensive assets to the region.
Most likely both sides would try and keep conflicts as localised and conventional weapon. Or at worse nuclear strike on advanced forces but not on territories of China/USA. Any victory in nuclear conflict is really only the matter of who has more people survive in radioactive wasteland they called home before. That's really a last resort weapon against invading enemy forces, but more so deterant to attempt to invade nuclear nation.
There are many other possible outcomes. But generally, USA will be reluctant to drop a bomb on capable country as opposed to being open to unleashing on incapable countries for transgressions.
Good example is Crimea: They would probably threaten on TV, enact some sanctions but ultimately won't do anything. It's doubtful USA would get involved in direct confrontation even if Russia invaded entire Ukraine and Baltic states. If it was let's say Romania instead of Russia, you'd be watching F22s on TV dropping freedom and democracy bombs in some cleverly named intervention operation. It's an easy way of looking at likelihood of conflict with China. China has a pretty big stick itself, like Russia.
