There's no reason the US will give up claim on California, and while Japan's main islands have moved, Okinawa is still right where it was, thus there's still forward basing to move support into Cali if required and to stage humanitarian efforts out of (Okinawa was used extensively by the US military to stage early relief for the Sendai Earthquake relief efforts).
That said, people discussing water I have good news: this switch actually is taking place right as the normal
Japanese rainy season begins, this means that much of California will see more rain in the next few months than it has ever. This will help alleviate, in the short term at least, the loss of the rivers and buy time for bringing online desalinization plants and moving population around.
Now, here's the thing, I don't see Japan deciding to politically integrate with the US in the short and even medium term. There's to many cultural, political, and legal differences between the two countries. However, the US and Japan already have a very good relationship, and I'd expect that relationship to only improve given the geographic proximity, believe it or not. Japan is already a country where citizens of the US and Japan can travel to freely for short term with just a passport, so aside from new connective roads needing to be built to integrate things (ones that would actually be complicated, considering that Japan is left side drive for... God knows why?), you'd probably see a lot of Americans and Japanese crossing the borders for vacations and tourism, leading to longer term close economic integration.
That said, we might well see the JSDF become integrated with the US Military much more than it already is with special joint commands, and the JSDF would definitely be integrated into NORAD and the other formerly just US/Canada joint military commands.
As to East Asia, suddenly China is going to be a lot more careful in how they handle activities in the regions. While S.Korea, Japan, and Taiwan were already in the US political and military sphere of influence, there's a difference between dealing with US... "allies" (*cough*vassals*cough*) and dealing with the US PROPER being there. North Korea; however, is now a wildcard that EVERYONE, including China, is going to be looking askance at. Remember, N. Korean missile tests have done overflies of Japan before, and since those were only threatening Japan the US, while it took them seriously, didn't see it as a personal threat to American citizens. Now though? The US has American citizens within nuclear range of a hostile foreign power that is NOT a fully rational actor. This will lead to considerable pressure from California to SOLVE the N.Korea issue definitively, though the exact HOW will depend on what kind of Californians remained in California and how it shifted politically due to its transference. This crisis could take a while to come to fruition, as the first few years will be spent mostly focused on the immediate problems and the longer term issues won't play out, and I suspect China will lean heavily on N.Korea to not rock the boat while America is in panic mode dealing with the immediate fallout, as when America is in panic mode it can act... irrationally and threatening the US with nuking California then is likely just to see the US lash out...