Good point. There is another that is sure as heck not unique to RCC or EO. I've even heard preachers tell how it's a symbol of God's eternity. But that, to those who know, is BS. It is a pagan symbol that has been adopted by Christians.
I've actually wondered about the wedding ring before, and did some research into it. To answer your question: I do not, my minister does.
Then your religion is "contaminated," too.
But I have a simpler explanation. Answer some easy Bible questions for me. Open your Bible, and tell me what happened under the Mosaic Law when an observant Jew touched a leper, a menstruating woman, or a dead body.
Then open the Gospel of Matthew and tell me what happened when Jesus touched each of those things.
Then reflect on what happens when his Church -- the Body of Christ, remember -- appropriates things that God created (he called them "good," remember?) and pagans hijacked, and restores them to the worship of the True God. What gets contaminated, and what gets healed?