We're not certain which insect larva they may be as the most common true maggot (a fly larva) would be Spotted-Wing Drosophila, which is a type of fruit fly. (The linked page is intended for agriculture instead of home gardeners, but the images and information about the fly and its habits still applies.) They're pretty small, though, so if you're seeing a larger larva, it might be Oriental Fruit Moth. If you see them again this coming season, feel free to send us photos for ID. Plum Curculio, a weevil (type of beetle) larva, also looks somewhat maggot-like in that it is grubby with no legs, and they can bore into fruit as well.

Peaches are vulnerable to a number of pest and disease issues, so benefit from annual preventative pesticide treatments (even if you opt for organic choices) or at least very close monitoring since few conditions are curable after their symptoms appear. You can learn more about peach tree care and troubleshooting in our linked web pages.

Miri

My recollection is that in each peach I saw 1 "worm" that was white and about 3/8 inches long

Without an image to assess it's hard to say which species it is, but at that size, it's too big for Spotted-Wing Drosophila, whose larvae only get about one-eighth of an inch long. Therefore, it may have been a caterpillar (the fruit moth mentioned) or a beetle grub (the curculio) instead of a fly maggot.

Miri

Any general measures I can take to prevent bugs?  ????

Yes, preventative pesticide sprays (even low-risk organic types) will help to nip pest problems in the bud. They need applying every year, and often multiple times per year, depending on the chemical and the pest or disease being targeted. The information on the pages we linked to previously present all of this detail, though we can link directly to the Virginia Tech fruit treatment publication directly: Home Fruit Disease and Pest Prevention Spray Schedule. You can also use our Fruit Problems: Prevention and Management page and the When to Spray Fruit Trees for examples of the different growth stages in spring to use as a timing guideline.

If you do not wish to use any pesticides, the only recourse might be to physically cover the developing fruits individually with mesh or paper bags. (An example by Clemson University can be seen in the linked short YouTube video. They had a web page about this item as well, but it appears to have a broken link at the moment.)

Miri