That article seems to go way off the deep end in the references. But in investigating this briefly to give some backup for that response, I find that the early church fathers did hold a similar view. Clement of Alexandria spoke strongly against them. Tertullian took them as having a clear pagan origin, but saw less harm in them if the focus was not on the pagan element. Basically, I did some background reading to show why the author was wrong, but my background reading didn't entirely support my initial hypothesis...
Rings appear to be strongly associated with authority and ownership. Historically, men would wear signet rings, with which they could seal their authority over particular items / letters. Such a signet ring could be given by a man in authority to his servant (e.g. the Pharaoh and Joseph) to delegate authority. Then, men also gave rings, originally iron, to their brides to signify their ownership of the wife. Neither usage is unscriptural.
Where the church fathers questioned the practice is where the ring becomes an outward ornament that is contrary to the instruction to dress modestly. A plain gold ring is not a particularly extravagent ornament in comparison to what we see around us, so that's not a critical issue in my mind in our particular culture.
The practice of both men and women giving each other rings, and both wearing them, is more questionable. It is very recent so not covered by early Christian writers. A man giving his wife a ring to symbolise that she belongs to him is entirely scriptural - both giving each other rings symbolises mutual submission that no longer represents the true patriarchal nature of biblical marriage, and is a very monogamous practice also. That I think is worth pondering more deeply, but for different reasons.
