I'm not going to let my homeless friend stay over anymore
Not really sure what you mean when you say stay over.
It means that I let him sleep over at my place for a night or so.
It's understandable wanting privacy and space in your own home but refusing a homeless friend shelter isn't something that is right. Friends help friends when they are in difficult circumstances. Rather than kick out your friend after a night or so, allow him to stay over a few weeks to allow him to get back on his feet without the constant worry of where he is going to sleep for the night.
I don't care about him. I just feel for him in a way.
You should still try to find a place where he can stay. While nobody is forcing you to let him stay with you, it seems wrong to just abandon him when he really needs someplace to crash.