The US doesn't lay claim to the Pacific Ocean, and it has a right to transit its navy in open waters. It does not make a habit of navigating its navy into other nation's waters without their permission.
The US does not claim ownership over any part of the Gulf of Mexico that it doesn't blatantly have a right to, as this image makes very clear:
What if China ran their war ships into Mexico gulf just like U.S did in Southern China Sea?We shall think it is quiet fair because in the "public" ocean?
That's exactly what would happen: the US would not freak out about it.
Russia was just busy flying bombers 40 miles off the coast of California last week. Last time I checked, we haven't declared war on Russia for it. They regularly do the same thing with their navy in various ways, whether off the Atlantic or Pacific coast, or near Cuba / Florida waters.
There's a pretty big difference in China sending a couple of warships to the gulf and USSR setting up a nuclear missile base with missiles pointed at the US on Cuba.
Nuclear missles or "90000 tons of diplomacy" are both certain kinds of deterrence, nothing more, nothing less. If US does not have those carrier combat groups, China wouldn't care either.
I'm going to sound like an apologist, but the Cold War nuclear miltary-political calculus was very different pre-ICMB+MAD on both sides of the Soviet-US split.
SLBMs were just being developed in the early 60s, and so Cuban-based missiles represented a novel, more credible first-strike threat given distance (~20 minutes flight time).
The US does not claim ownership over any part of the Gulf of Mexico that it doesn't blatantly have a right to, as this image makes very clear:
https://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Te...
The southern tip of Texas across to the southern tip of Florida is extremely reasonable as a water claim.