This is a really interesting discussion! There are just a couple of things I want to add to what's already been said:
The suggestion that Edward is an atheist is odd, to say the least. Are we reading the same books? If Edward didn't believe in God he wouldn't be concerned about souls and doing things right to avoid damnation, because atheists don't believe there's a God out there to do any damning. He would probably still be opposed to changing Bella because it's not morally right to kill someone, even if that someone is going to go on living as a vampire, but all his arguments about heaven and hell, her soul, etc, would be irrelevant to an atheist. Edward believes in God, he just doesn't believe - or doesn't want to let himself believe - that God could want anything to do with a vampire.
Someone asked why Carlisle, at least, if not the others, didn't continue to go to church after they became vampires. First of all, I don't think there's any indication that they don't - Bella never explicitly says that "It was Sunday morning so Carlisle and Esme were at church", but since she doesn't attend church herself she wouldn't have known if they did before she became part of their family, and she leaves out a lot of the personal details and schedules of the Cullens who aren't Edward, so it's possible she just never found it interesting enough to mention.
Assuming none of them do attend church, though (which I think is more likely), I doubt it's out of any fear that they'd burn up on entering holy ground or anything like that; Carlisle probably even tried that as one of his suicide methods when he first became a vampire. Besides, Rosalie and Emmett have had quite a few weddings, and you can't tell me Rosalie wouldn't have wanted at least one in the most beautiful cathedral she could find!
I think it's more likely that they just aren't sure what the rules are anymore, and so aren't going to find much solace in a church service. Carlisle has some hope that by living a peaceful life and resisting their murderous impulses they might be able to get into heaven if they ever die, but even he isn't sure, and so the things human believers can take comfort in - forgiveness of sins, the promise of a better world after this life - would be more like rubbing salt in a wound. There's also the fact that, barring a careless moment in battle or something similar, death isn't something vampires have to worry about much. For all intents and purposes, they'll live forever. Why waste time in a church being told that when you die you can go to heaven if death isn't something you're likely to experience?
(ETA: I'm not saying church is a waste of time! I'm a Christian; I don't think that at all! But for a vampire, it kind of would be.)