TBH I don't understand what's the issue here.
You don't mention if you or your friends are male or female (or other), but as a gay man I wouldn't have any issue with someone saying they wished they were gay.
Surely it's a compliment not something that they could hate you over?
Are you possibly over-thinking this?

I guess I could be overthinking but I just feel bad because I feel like without an explanation it kind of sounds like I’m saying “I want less rights” and I don’t want to be insensitive.

What did you do that was wrong? I don't see anything.

Well some people may not want to be gay because of religion or their family is against it or from the lack of rights. I don’t want to seem like I’m completely ignoring that.

You don't mention their reaction to your statement. If they laughed or the conversation kept on in the same tone, I would assume no one was offended and they all took your joke in the manner it was intended. No harm no foul.

One of them pointed it out and laughed, but I was a bit too embarrassed to look at anyone until the conversation moved on.

I'm gay.. and you said or did nothing wrong.

Stop overthinking it.

Pointed it out like mentioned the same flaw in your statement that you did in your post? If that's what happened, then they all processed that you said something mildly dumb (like seriously, it was a very mildly thoughtless statement. You weren't being insulting, just not thinking of the implications. Everybody does it sometimes), saw that you were embarrassed, and moved on. I'm assuming you wouldn't hold a similar statement against one of your guy friends ("I wish I was a woman so I could wear dresses!" "Really? Not sure a cute miniskirt is worth the sexual harassment and being passed over for promotions at work but go on."), right? You learned a lesson about thinking before speaking; there's no point in continually beating yourself up over it, just let the experience be your guide for the future. That's how people grow. Your friends don't hate you and probably forgot about it 5 min after it happened.