If you want to avoid sprays, you can use fruit protection bags to protect against most insects, but you need to bag all of the fruit you want to protect. These bags are readily available online and may be available locally. The trick is that early pests arrive shortly after petal fall, so for best protection you need to apply the bags around flower clusters once most of the petals have dropped. You will then need to remove those bags and thin the fruit once the fruit is about 3/4-inch in diameter. Thin the peaches to 8-inches apart, which is about the breadth of the hand from tip of the pinkie to tip of the thumb when the fingers are splayed. This will help protect the branches from breaking under excess weight of fruit, and it will ensure the peaches are large and high quality. 

Also keep an eye for evidence of peach tree borers. Usually it shows up as gummy sap exuding from a hole near the base of the trunk.

For more information:

Growing peaches in the backyard

Peach tree borer

https://www.extension.purdue.edu/extmedia/ID/ID-146-W.pdf

https://s3.wp.wsu.edu/uploads/sites/2070/2019/09/EM066E.pdf