> China says it's their territory, the US says it is international waters. And so we "prove" it by regularly traveling through it with US warships for no other purpose then to prove we can.
Suppose the US claimed that the entire Gulf of Mexico was US territory. Would people be as sympathetic to that as they appear to be to China claiming that the entire South China Sea is Chinese territory? (Note that the South China Sea is more than twice as large as the Gulf of Mexico, and less of its coastline is part of China as compared with the amount of Gulf of Mexico coastline that is part of the US.)
> Since China is a long way from the Gulf of Mexico, and has no need to sail there, I'm confident that China couldn't care less about this.
First, many Chinese flag ships sail through the Gulf of Mexico, so I think China does care.
Second, even if China didn't care, many other countries do. And the US, unlike China, respects the rights of other countries to freedom of the seas.
> The South China Sea is very important to China, and very far away from USA -- why does USA care so much about it?
First, because many US flag ships sail through the South China Sea. We trade with many countries other than China that have coastlines on that sea. You know that, right?
Second, because, just as with the Gulf of Mexico, many other countries have ships that sail through the South China Sea, and there are internationally agreed rules about international waters and freedom of the seas, which, as noted above, China does not respect the way the US does. The only way to enforce such rules is for countries that uphold them to take action against countries that do not.
> Perhaps you could also comment of USA claims over the Northern Sea Route (which lies within Russia’s Exclusive Economic Zone) ?
This case is an excellent illustration of the proper way to negotiate rules regarding freedom of the seas: to notify other countries of something you would like to see happen, and then have a diplomatic dialogue with them. As opposed to, oh, say, just asserting that a huge chunk of ocean is your territorial waters, in contravention of international law and treaties.
Also, since you mentioned Exclusive Economic Zones, you will note that the US is not saying it wants to exploit resources in this zone; it is only saying it wants it to be a transport corridor. Which, according to current international agreements, it already is everywhere outside 12 nautical miles from Russia's coast.
Suppose the US claimed that the entire Gulf of Mexico was US territory. Would people be as sympathetic to that as they appear to be to China claiming that the entire South China Sea is Chinese territory? (Note that the South China Sea is more than twice as large as the Gulf of Mexico, and less of its coastline is part of China as compared with the amount of Gulf of Mexico coastline that is part of the US.)