Peel, pit, and cut peaches into 1/2-inch chunks; set aside. Heat oil in a small saucepan over medium heat. Cook onion until soft, 5 minutes. Add garlic and red-pepper flakes; cook 1 minute.
Stir in vinegar, sugar, and cherries; season with salt and pepper. Bring to a boil; reduce heat, and simmer until syrupy, about 5 minutes.
Add peaches; simmer until soft, 5 to 10 minutes. With a wooden spoon, mash some peaches against side of pan; thin with 2 to 4 tablespoons water, if needed. Cool completely. Serve with grilled meat or poultry.