I don't think a law determines the morality of a situation like this, a persons conscience does. If you can live with your decision then do that. I can understand your view AND the view of your wife. It doesn't mean either of you are wrong. What it means is that maybe you should keep your ring finds to yourself. It would probably be my decision if I was in your situation.
I would feel no more obligated to tell my wife I found a ring while enjoying my hobby (if I knew it would cause this kind of situation) than I would telling her I missed a jump shot if basketball was my hobby.
If you feel morally obligated to tell her....Well then, you are really going to have a lot of explaining to do in this hobby.
I am in the camp that if YOU can find the rightful owner then do so. Otherwise it's yours. If you go asking if anyone lost a gold ring I am sure that you can always find those that will say it's theirs when it isn't... I would look at that as me doing something that could possibly cause someone else to be immoral (IMO) and well I would have a hard time living with that.