Not our problem.
Unfortunately, you're wrong. The Treaty of Mutual Cooperation and Security between the United States and Japan requires the US to intervene on Japan's side in military conflicts. So it is our problem, even though no American should have to die for a couple of worthless uninhabited rocks. There are also about 50,000 American military personnel in Japan at any one time, sitting ducks in any war between the two countries, including the US Seventh Fleet.