My wife wants me to. So I wear it. Take it off in the gym and that's about it. Doesn't really bother me to wear it or not. So if she feels like she's winning because of it, great. I hum and haw about it, but always give in.
Negotiation (and hence marriage, which is just a lifelong negotiation) is all about giving in to the other party on points of contention that are a big deal for them, yet aren't for you...and then playing up how much of a big deal it was for you to give in on that one (albeit inconsequential to you) point. I then use the principle of "can't say no twice in a row" to get what I really want.
Wife: "Hey, how come you're going out with the guys to the pub again? And where's your wedding ring?"
Me: "Aw, come on. I never see the guys anymore since Mike had a kid. It's one time. And why do you care if I wear my ring...are you afraid some girl at the bar will hit on me?" (clever question...if she answers in the affirmative, she displays weakness and beta female tendencies. If she answers in the negative, she admits beta tendencies for marrying an unattractive sod. I know she will deflect)
Wife: "Well, I just wish you'd wear it, is all. Don't you like it?" (here's her misdirection, into a full on assault. If I answer in the negative, I cast aspersions on our entire union...the ring being a reflection of her and our marriage. Thus, the only wise answer is in the affirmative, but with a reciprocal ask of my own)
Me: "Of course I love the ring. It's beautiful. You know, I didn't know you felt so strongly about it. Of course I'll wear it out tonight at the pub with the guys. I love you!" (of course I know she feels strongly about it...we argue about this at least once per month. But I close the statement off with an "I love you", which women love to hear and improves her security. And I also close it off by letting her know how proud I am to proclaim our union at the pub tonight - again a security blanket).
Wife: "Ok sweetie, don't be too late. Wake me up when you get home."
Result for her: She gets me to wear my wedding band in public, secure in the belief that any woman who lays eyes on me will run screaming for the hills due to the wedding band, even though I probably would have worn the thing anyways
Result for me: I pretend to not want to wear it, to set up a reciprocal give-and-take situation on the issue of the ring in order to score a free guilt-free night out at the pub with friends.