The peach you ate was a hybrid so likely you wil get no peaches from any kind of shoot it sends up. You will get a "tree" but no peaches.
Hybrids aren't by nature sterile. Every domestic animal and every domestic crop is a hybrid. Sometimes a hybrid can be sterile, such as mules.
I believe you are mixing up a variety with a hybrid. A Aprium is an example of a plum/apricot hybrid, but most peaches are cultivars or varieties, and as such their seedlings will not be always be true to variety, but will usually bare peach or sometimes nectarine (fuzzless peach)fruit.