Preheat oven to 425 degrees. Drain spinach; transfer to a large bowl.
Add ricotta, garlic, 1 teaspoon salt, and 1/4 teaspoon pepper; stir well to combine. Set filling aside.
Place a chicken breast on a clean work surface. With your fingers, loosen the skin; place a quarter of the filling underneath the skin of the breast. With one hand on top of chicken and the other keeping opening closed, press filling evenly to edges. Repeat with remaining breasts and filling. Season chicken breasts with salt and pepper.
Place on a rimmed baking sheet; roast until chicken is cooked through, 25 to 30 minutes. If you like, scatter the tomatoes on the same baking sheet around the chicken. Serve immediately.