I don't think he will, not as a practical matter.
In the modern era, I can't think of a single, major-party candidate - and certainly not an incumbent - who wasn't on the ballot in all 57 states ;)
If Obama is disallowed ballot access in a state, even AZ, because of his refusal to provide the appropriate documentation, that will be a HUGE story. In fact, I would say it would be the biggest election story in generations. He can't, as a practical political matter, ignore it.
The irony is he would be forced to be the plaintiff in the case that might (although unlikely) disqualify him from office.
Let’s hope you’re correct. I will feel a lot more confident though if FL challenges him as AZ has.