Technically, he can quit-claim the deed to you without you doing a refi. He just has no incentive to do so. Normally the quit-claim is your incentive to get the refi in a reasonable amount of time.
I quit-claimed the house to my X and did not require a refi. My name is still on the mortgage. There is a "hold harmless" clause in the Decree which says she will reimburse me (out of her alimony) if the bank comes after me for the mortgage. But it still does a number on my debt ratio. Obviously this scenario wouldn't be appealing to your H--what you've already got is more the norm.
