The cause of the problem is simple.
Christians choose to be Christian, and Muslims choose to be Muslim. Thus, they choose to be evil on the topic of homosexuality.
First, this is black and white thinking of the same sort they engage in. Reality isn't that neat and tidy; and belief is not something one actually has much conscious choice in. Suggesting otherwise makes it seem as if we choose the way we see the universe we live in the same manner one would choose whether to have a vanilla or a chocolate milkshake. We simply don't have that kind of control over how we see the world and those within it. Indeed, if we did, the world wouldn't be as fucked up as it is. You can choose to walk into a church, and that might lead you down a path that eventually has you believe in the christian religion... but one doesn't simply choose to, from one moment to the next, to be christian. Just as one doesn't simply choose to be evil.
Secondly, religion itself obviously isn't the real cause and you're doing both the bigots and their victims a great disservice by using this sort of rhetoric.
"Who cares why the enemy is our enemy? All that matters is that they are the enemy."
History shows that if we point out their evil often enough, they will reform and insist that their religion was always against the persecution of homosexuals, but they will only do this if we point out that their teachings and actions based on those teachings are evil.
You and I must read some very different history books.
Besides, what you're talking about *never* happens as a result of telling people that their teachings and actions are evil. You don't seriously believe that going up to a rapist and telling him "Hey, rape is evil!" is going to make him change his ways, do you? Why would you think that's how it works with religion? No, people don't change their ways by actively going up to to tell them they're evil. That just annoys them and makes them defensive. All you'd accomplish is for them to dig their heels deeper into the sand.
Hatred thrives on conflict; which is what you're giving it. This is why largely ignoring them is the only viable long-term strategy. Modern society didn't get to be as tolerant as it is by constantly moralizing the old-fashioned bigots; it got there by allowing them to gently turn irrelevant. We occasionally mock them, and yes point out that this or that is bad... but pushing it too hard will always backfire.
