It is difficult to say with exact certainty, but it could be the peach twig borer.  This is a serious pest of stone fruits in Utah.  Utah State has a great Fact Sheet on it: http://extension.usu.edu/files/publications/factsheet/peach-twig-borers07.pdf

While it mostly affects twigs, it can affect maturing fruit in late summer.  Controls vary and you can see the recommendations...mostly insecticidal in early spring, but some cultural controls as well.

Have you dug into the peaches to see if you can find any larvae?  Have you noticed any adult insects on the tree?  Have you changed any management practices?  Cultural conditions?

If you can find adults and/or larvae, that would help.  The Utah State link has excellent photos that may confirm/unconfirm the diagnosis.

Thank you for your reply. I looked at the link and don't think this is the culprit. I haven't found any insects on the tree, no deformed foliage, and no larvae in or on the peaches. This is a backyard tree that just kind of does its thing with a little supplemental watering. It is the only one (of 2) that produced this year due to late spring snow. It would have been interesting to see if both trees (not near each other) would have had the same issue. I also have a nectarine which seems fine,although the fruit is still very unripe. 
I remain stumped.

One thing you could consider is visiting a farmers' market and talking to a peach grower.  At the Larimer County Farmers' Market in Old Town (at the courthouse), we have a couple peach growers.  Either Steve Ela with Ela Family Farms would be great to talk to...or Dan Collette with Collette Fruit Orchards.  They are in the business and may have recommendations on what could be the culprit.

As you can expect, it's very difficult to diagnose any insect or disease issues via email :)

Thanks again. As I started cutting peaches today, I actually found several larvae in the rotted areas. They are in different stages of development. My husband (an entomologist himself) is doing some investigating.