Still, as suggested in the post above, Levi may be feeling some pangs of regret as he gets older, with respect to his son, and Bristol may have indeed made the apologies a condition of greater access. I can see that happening, and I can see also her still wanting Tripp's real father around as a positive thing.
But, before he can re-earn the the Palins' trust, he'll need to publicly disown every one of his recent liberal associates, get his GED, and then probably join the military to make something out of himself so that he can properly support Bristol and Tripp.
I don’t see him regaining their trust, just getting out of the way by removing himself as a useful tool for the media that hates her.
His silence will suffice. Remember, he lost in court to Bristol. He’s lost every battle he’s waged, save for the media trashing of the Palin’s and Sarah in particular of course. If this his his media swan song, then he exits on a good note and both he and Sarah win, while the sorry media and her detractors LOSE.
Levi belongs in Alaska as a private citizen with nothing more to say to the media for the rest of his life.