Oh, just to make the little issue that millions in the country doubt his eligibility go away. Must not be important to you?
First of all, I doubt the number of people who care about this issue is in the millions.
Second of all, it wouldn't go away no matter what he showed them. If someone showed a judge the vault copy of his BC, the birthers would find some way to argue the judge was in on the conspiracy and not to be trusted. If he posted an image of the vault copy on the Internet, some anonymous guy with a phony Ph.D. would write a 200 page treatise claiming to have "conclusive proof" it was a forgery.
Even if he could, by some miracle, convince the entire birther fringe that the BC was geninue, they'd be filing lawsuits claiming he is ineligible because his Father was Kenyan, or because he lived in Indonesia, or some other half-baked nonsense.
The simple answer as to why he hasn't done more to satisfy the birthers is there's no satisfying them.