In so far as we are aware, no one has ever questioned Romney's birth in Michigan, nor is there any evidence to suggest that he wasn't born there as claimed. If would seem like a waste of effort for a political opponent to go to any great lengths to verify his birth in the state, because it's nearly a certainty.
And as for Romney himself at this point in the presidential campaign, he would be reluctant to display his birth certificate publicly, simply because opponents would accuse him of playing up the "birther" issue against Obama if he did so.