I am not aware of an appellate decision or statute in Utah which sets out the law for this question. If there is another attorney out there who does, I would encourage him or her to write a response. But it just might be that no law exists.
Without clear law, most courts will probably look at this as either a contract issue, or a gift issue.
If you paid for the ring, and it cost less than $10,000, you might sue in Small Claims for the value of the ring using a contract theory. However, if the ring cost more than $10,000 you would need to file in District Court.
Using a contract theory, you would argue that the ring was given to your fiancé for the sole purpose of your fiancé completing a marriage contract with you. Since the marriage never occurred, and the marriage contract was therefore never concluded, you are entitled to recover the value of the ring.
If the ring is one of a kind (for example, if it's a handmade ring that has been passed down generation after generation in your family, and is a one of a kind heirloom), then you might sue in District Court for the actual return of the ring, no matter the value. Small Claims courts can only award money judgments.
But it's rare to have a court order the return of personal property. Even District Courts almost always award a money judgment instead. Having said that, the law does allow a District Court to order that the property be returned if the item you seek is unique, one of a kind, and can't be replaced.
If you are the one receiving the ring and your fiancé sues you, you might argue that the ring is a gift . Depending on the facts, your fiancé might try to say that you never intended to marry in the first place, and therefore committed fraud in inducing him to purchase and give you the ring. It becomes a he said, she said, trial and the court would need to decide who to believe. The problem, of course, is that most people who date and then decide to marry don't do it in an absolutely straightforward, clear and concise manner. As we all know, romance is messy. And so are the factual situations that flow out of romance.
If instead of a gift, the court finds that receipt of the ring by you was part of a contract, you could argue that you fully performed your end of the contract when you said that you would marry. The ring was offered as an inducement for your agreement to marry, and you agreed. The fact that the marriage ultimately didn't happen should not matter. On the day you said yes, the day the ring was given to you, you were ready, able, and willing to marry. The contract to receive the ring was therefore complete.
Ultimately, though, I believe most courts will do their best to put you both back in the position you were in before the ring exchanged hands. Whichever fiancé purchased the ring will be given a money judgment against the other fiancé for the value of the ring. Unless, of course, the ring is returned.
But there could be other outcomes, depending on the facts of the case.
If this is an issue you are serious about, I recommend you contact an attorney and have that attorney research the issue after an interview with you so that the attorney learns all of the facts.
I hope this helps.