The style of the ring or the quality of manufacture might be characteristic of a particular historic period. An expert might be able to distinguish a Tudor era ring from an Anglo Saxon era ring from a Roman era ring.

It's conceivable that if the ring has a stone set in it, perhaps there's a bit of wax or resin used to hold the stone in place while the setting is adjusted. If there were, it might be possible to carbon date it.

The gold itself of course is ancient, probably older than Earth itself.

If you melt down a Tudor ring and make a new ring, or even just hammer the ring into a new shape, it won't be a Tudor ring any more and there won't be any evidence it ever was one.