Fill a medium bowl with ice water. Submerge onion in ice water, and let stand 10 minutes. Drain and pat onion dry.
Meanwhile, stir together yogurt, chopped lemon herbs, honey, 2 teaspoons lemon juice, and 1/2 teaspoon salt in a medium bowl until blended; set aside.
Gently fold together watermelon, cantaloupe, garlic oil, chile, remaining 1 tablespoon lemon juice, and remaining 2 teaspoons salt in a large bowl.
To serve, spread yogurt mixture evenly on a large serving platter. Using a slotted spoon, top with melon mixture. Sprinkle with Aleppo, and top with onion slices. Grate lime over melon mixture, and garnish with lemon herb leaves.