Combine diced melon, 1/4 cup chopped cilantro, finely diced red onion, lime juice, olive oil, grated lime peel, and chopped serrano chile in medium bowl. Toss to blend flavors. Season melon salsa to taste with salt and pepper.
Spray grill with nonstick vegetable oil spray; prepare barbecue (medium heat). Sprinkle chicken breasts with salt and pepper. Grill chicken breasts until skin is crisp and brown and chicken is cooked through, about 5 minutes per side.
Slice chicken breasts and arrange on plates. Top with melon salsa and cilantro leaves and serve.