I think that "being gay is a choice" is a phrase that Christianity owns.
Judaism has no real particular view on it, even though there are Jews who are gay, they do seek out help with that.
Think of it this way. You may not choose who you're attracted to, but you DO choose every relationship you enter for any purpose, regardless of sex being involved or not. When the consequences are spelled out beforehand, you can not say you weren't informed.
Also remember that heterosexual fornication is equally a sin in God's eye; unmarried heterosexual fornicators will face the exact same judgment.
Not all religious people are that ignorant. There are many in all three faiths who know it isn't a choice. And there are only two branches of Judaism and a few branches of Christianity who think it is a sin. (Along with Islam, although many individual Muslims disagree.)
The rest of Judaism and Christianity accept homosexuality, since the Hebrew and Greek texts of scripture don't condemn it.
Yes it's a choice from the perspective that, you can choose to obey God's law - which is that homosexuality is a sin and so whatever they crave to do, they will not do it because it's against God's commands. It's obedience and faith which pleases God.