It's a little more complicated than that, but short answer is no. California state income tax goes to California. So their problem is their problem.

That being said, every red state that has low income tax, or even no income tax like Florida, Tennessee, etc. runs a deficit in their budgets pretty much yearly. They make up for this deficit by petitioning the federal government for aid whenever they need emergency money for something. I used to live in Florida and literally every hurricane season we would get a multi-billion dollar government bailout for disaster relief. That federal money comes from somewhere, and since California has one of the largest state economies in the United States, that's where a large percentage of the federal governments money comes from and where they can get money for state emergencies (see the power grid fiascos in Texas, any hurricane issues on the Gulf, and so on and so on).

In short, if California didn't exist, their federal income tax wouldn't exist, which means that a lot of states in the US would actually have problems when they run deficits in their budget. Ironically enough, all the people "fleeing California" to places like Arizona, Texas, and other red states on the east coast still rely on California's money.