It's possible (though I'd say not likely- come on now) that he was only looking on the gay hookup site as he says- repeatedly.
If so, then he should have no problem giving you full access to that account and his phone and computer, too. I think you'd be fully justified to demand that, given the circumstances.
But it's probably more likely that he will not do that and honestly that you're no more eager to hear that type of truth than he is to say it. I say this because you seemed to accept his words that he wanted to stay with you and be faithful quite easily, in spite of all that had just happened.
I think most likely he's been cheating on you with men for some time. And now that some of the truth he's been hiding has accidentally been found out, he's scrambling to hold it all together. What he most likely wants to do, in my opinion, is to continue to have his cake and eat it too.
If you look the other way on this, you likely ARE in an open relationship, on one end anyway, you just don't know it. Personally, I'd much rather figure out some kind of open relationship rather than accept lies, cheating, the chance of getting an STI and the chance of being suddenly left when and if he gets something set up with someone else.
To me, he'd have to earn his way back in. He's kept enough from you and given you quite reasonable reason for doubt. Just saying he wants everything to continue would not be for me. Hope it all works out for you.