It is peach tree sap. It might be a bad news... Tree has an open wound trying to seal itself which means you might have a Peach tree borer usually on the trunk, or other peach tree diseases.

The peaches are oozing gum. The most common reason is you have oriental fruit moth in the fruits. Peaches with plum curculio also can ooze, but not big quantities like OFM damaged peaches. Cut open some of the oozing fruits and look for a white worm and a trail inside. Sometimes peaches ooze because there is a physical injury or problem with how the fruit grew, and lots of rain can cause them to ooze in those spots. The stuff on the leaves probably fell off some fruits.

Scott