Whisk together lime juice, sugar, ginger, and oil; season with coarse salt.
Preheat grill to medium. Thread swordfish and scallion whites onto skewers. Brush fish and scallions with oil and season with coarse salt and pepper. Grill, flipping occasionally, until fish is opaque throughout, about 7 minutes.
Divide lettuce and watermelon among plates. Top with skewers. Drizzle with dressing and top with scallion greens and carrot. Sprinkle with sea salt and serve immediately.