Probably not, but recalls of governors are rather rare. I also note that it's somewhat duplicative for a Democratic governor of CA to run for President in that CA is going Dem anyway. They'd be better off hailing from Texas or Ohio or even Georgia if they are reasonably popular at home.
I think he's looking at a run, depending, and will become more visible on "issues" like this one over the next couple of years. He might even be seen doing his own laundry somewhere not French.