For safety? Yeah, he does. Gyms vary on how fussy they are about it, but at the very least you can wreck your ring (or your hand) pretty badly, so it's a good idea to take it off.
That said, the real problem here is that your man taking off his ring makes you worried. Ask yourself: why do you feel this way? Is it something he did? Is it from something that happened to you before? Is it because of other ways he's acting?
If your man is making you paranoid, talk to him about it. If nothing else, you'll feel better it's out in the open.
Also your hands can get quite sweaty at the gym, and that can lead to ring slippage, better he put it in his shoe/bag in a locker than have it go flying and go down a grate or something equally unpleasant, right?
I do parkour and I always take my ring off. I forgot once and scratched it rather badly. nbd.