I doubt if you will succeed but it costs nothing to try. I can only guess that your peach is in bloom or has leafed out already. This is not the best time to take cuttings. Al

Peach needs to be grafted onto good rootstock or useless, even if you can get roots.

It can be done. I've rooted thousands, over the years. It's, however, probably a little late in the season for it. The best time is before they have leafed out. The time around here (NYC area) is around St. Patrick's Day. This, conveniently, is also when peach trees are pruned so, if you have a commercial peach grower in your area, you can probably get some of the prunings.

You want to use cuttings that are about the diameter, and length, of a wood pencil. Dip the cutting in hormone, and stick in a mixture of half peat and half perlite, in a bed, outside, with bottom heat. The idea is to keep the bottoms warm, and the tops cool, so you get root development, before too much leafing out.

Grafting of fruit trees onto different root stocks is done for several reasons. A standard apple tree usually gets too large for most people, so they graft them onto dwarfing root stocks. However, peaches don't get that large, so the main reasons for different root stocks are things like nematode resistance, etc. If you don't have a nematode or other problem in your area, rooted cuttings are perfectly fine.

Many old ideas die slowly. Peaches, and many other fruits, do not come reliably true from seed so, years ago, nurseries would get peach pits from canneries, and plant them, and then graft the desired variety onto the seedlings.

Now, we have better hormones, and they can even be cloned.

A couple of years ago, I stuck some cuttings in March. I was potting them up in June, and two, for some reason, got left in the bed. I didn't notice until September and, at that time, they were 3 feet tall, and one was even branched. The root balls were too big to fit in 1 gallon pots. The one that was branched, flowered, and produced fruit the next year.

Does anyone have great peach cuttings that they can mail me?
Also interested in delicious cherries, plums cranberries
And blueberries.
Will gladly pay ailing costs and even trade cuttings in the coming fall.
Just starting my fruit garden now. Also interested in hardy roses, figs and tasty grapes.
Thanks
Emprised@gmail.com

Just for the record a UF Beauty is a patented variety by University of Florida. I'm not certain if there would be liability for a private owner to propogate it for his own uses, but I'd proceed with caution :).

Here is a link that might be useful: Patent info