I'm guessing this was a misunderstanding. You were looking at him because he was looking at you. And he was looking at you because you were looking at him. You thought he was interested, and you don't mind that, so you gave him a show. But that's not what he wanted, in fact it probably confirmed his worst fears about you, and he was creeped out.

(Or maybe he did want a show, felt confused and upset about it later, and decided to shift the blame to you in order to try to make himself feel better about his own reaction.)

I think the lesson to be learned here is that if someone is looking at you, you can't assume it means that they are interested. Sometimes the only reason a person is looking at you is because they noticed you looking at them, and they're just trying to figure you out. Personally, I've been on both sides of that sort of misunderstanding, but mature adults ought to be able to say, "Sorry I misunderstood" and "Hey, no hard feelings" then let it go and move on.

I do think the guy overreacted by confronting you the way he did. He sounds insecure and seems to have anger management issues. Clearly he's a bit homophobic. The sad thing is, you're probably not the only guy who has freaked him out a bit. I wouldn't be surprised if he's one of those guys who complains to his friends about how "fags" hit on him all the time, because he thinks every time another guy looks at him, it's a come on. It's pretty typical of homophobes to have that sort of paranoid perception that other guys are sexually objectifying them, which is sometimes just a projection of their own feelings.

Unfortunately, by being so attentive to other guys and how they think other guys are perceiving them, homophobes often put out mixed signals, thereby drawing the exact sort of attention from other guys that they don't want (and perhaps unconsciously do want), resulting in misunderstandings like this one.

I think it's best to avoid this guy, but don't think you should go out of your way to do so. Go to the gym whenever is most convenient for you, and if he's there he's there, and if he's not he's not. If he is there, stay as far away from him as possible, try not to look at him, and don't give him any reason to complain about you. Good luck.