This is a shooting war between two nuclear powers. Either peace comes quickly or the worst will come to pass.
The UN could not stop the violence, given the power of the security council.
A engagement in the Taiwan straits is prone to escalate further--fighting on the interKorean border, US Air Attacks on PRC military targets.
The USA will use tactical nuclear weapons to stop a PRC/NK wipeout in Korea; from that point a general nuclear exchange is imminent. It's an exchange that will devastate the USA and destroy China entirely, and this gets into the post-Nuclear trauma that is bound to be bad for the world in a multitude of ways.
Few things say stupid more loudly than a shooting war between two nuclear powers. This case is quite possible if either side decides to reject negotiations.
LA has the odd quirk of ignoring nuclear war as an outcome, so I'm going to shelve it, in spite of the fact that "Neither Side willing to back down" would end in the devastation of the United States and the annihilation of the PRC.
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From the USA's point of view, the status quo is a victory. Taiwan's independence or at least its outright defiance is China's loss. This conflict is clearly China's to lose. The Independence CAG could get sunk, but what does this achieve? It essentially only extends a hopeless war for the PRC and diminishes the chance of a white peace.
Further Escalation would end in a nuclear war. The United States Economy hugely outpowers its own, to say nothing of the implications of endless bombing raids from the United States. Taiwan is the PRC's only possible gain in this war, but it is a war they can't win because the United States will almost certainly be able to retake the island, and several others, and then bomb the PRC silly.
So the PRC is humiliated. They are forced to back down in the face of the USA's Superior economic and military power. It is hard to know what concessions the PRC would concede, but claims on disputed islands between the PRC and Taiwan and formal recognition of Taiwan would probably be in the conditions.
The postwar world is going to depend dramatically on the nature and intensity of the fighting. Unless the PRC is depopulated in a nuclear holocaust, the USA is going to view it as a cold war continued scenario. The considerable US/PRC economic ties that emerge in OTL will not in this situation. There may also be much more rapprochement with Russia as a counterbalance in Central Asia.
The PRC government may well fall as well. Democracy protests in 1989 were crushed with force. How a PRC that played tough and is forced to back down at home fares is up in the air. The course of OTL--economic progress through access to global markets--will be closed to them. The PRC may collapse in another serious round of democratic protests, or it may find itself turning into a hawkish military state with even lower prospects than the Soviet Union in a second cold war.
The United States would almost certainly build another alliance club to deal with China. It will prevail, given time, but things will suck again for much of the world.