I'm worried that if your neighbour can resort to pasting rude notes on your door and passing hurting comments, he might go to the extent of harming you physically. I'm not trying to scare you, but some people can't even mind their own business and waste their time and energy focusing on such stuff.
Do you think you can afford to move out to another apartment? Maybe one that is LGBT friendly. Think about it and hopefully you can make a right decision.
One thing you can do is to ignore him. Its not easy but you can train yourself to do so. If he starts his nonsensical remarks, just have a thought in your mind that "Oh that idiot has started his nonsense but i dun give a damn about it." and ignore him.
Just think that his mouth needs to break wind because his ass has a problem. ( sorry if i sound vulgar. )
Or you could listen to your mp3 player while leaving your apartment. It might help to divert your attention. You can keep away your mp3 player after reaching outside.
As for the notes, you can safekeep them if you intend to approach a lawyer like what other members have suggested.
If you do not intend to, you can just spit on the notes and throw them away in the dustbin. That rude note is not even worth your spit man. ( Again sry for the vulgar content. )
I would like to end with a common internet phrase " There are over 7 billion people on earth. And you're going to let 1 person ruin your day?
"
Hope you can get this problem solved and lead a happy life.