Claims back to 15th century / US administration after WW2
Israel has taught us that it does not matter who owns what. If you and your ancestors have been living in the "wrong place" for the last thousand years, and some fucker comes along and says "Bad luck for you, these lands are now owned by someone else", you can get all angry and you can go Hamas, but they're still going to take away everything you rightfully owned for centuries, everything you worked for. And the international community will watch and be silent. A piece of paper will be written with some twisted words that say "new people own this land now", and so everything is legal. 20 years later, nobody will remember. All tha will be remembered is that
you are the bad terrorist guy.
It's not like it was any different for the pre-19th-century inhabitants of the USA who were driven off their lands and murdered in millions. Nobody gives a crap about them, they're gone. And it's all legal.
That's just how things go. You object to someone taking way your lands, you throw a rock at them, and they mow your people down in dozens with machine guns, and all that happens is that Amnesty International says "well folks, that wasn't very nice of you". Life goes on, but without you.
Will there be war?
No, but quite possibly China will just take over those isles one day.
Our People Needs Room -- Adolf Schicklgruber, 1939. The same is dramatically true for China, too. They have everything (in particular people), except room. It will become unavoidable that they acquire new lands.
Firstly America can’t afford [...]they lose financially if they fight China.
That's almost correct, but you got it on the wrong angle. It's not about the USA losing some money in an economically unattractive war. Fact is that China can anihilate the USA from one day to another if they throw their US bonds on the market and sell their dollar certificates. Will they lose a few billions? Yes, but who cares, China has enough money. They need not fire one nuclear missle to completely destroy the USA in one strike.
And that is why the USA will just keep it shut in case China decides to take over those islands. Maybe the foreign minister will make a statement about how shocked he is, then China will make a few strong words and play their muscles, and everyone will back off.
Let's not mention that the "not quite up to par" Chinese military is frightening at best. The sheer number of people and births they have would make China an easy winner in a non-nuclear-non-financial shootout as well. Yes, Wikipedia suggests something different, but if you look at those numbers for a seconds, you can only agree how ridiculous they are (the USA have 14 times as many people reaching military age per year, got to be kidding?).
North Korea
Now there's that, you can never predict what a total nutter will do. Kim might still just attack Japan (or even the USA) under this pretext -- although I doubt it. Not even Kim is that insane, and he must be able to figure that the expected losses are much greater than any possible gains for him.