In a small bowl whisk together lime, vinegar and sugar. Pour 1/2 of the dressing in a small container and set aside. Place onions in a shallow dish, add remaining dressing, and let stand 30 minutes or overnight.
Season fish with salt and pepper and drizzle with 1 teaspoon of the olive oil. Place the fish, skin side down, in a non-stick pan with an oven proof handle. Bake until fish is cooked through, about 12 to 15 minutes. Cut fish into 4 pieces and place 1 piece on each serving dish.
Place watercress, arugula and melon in a large salad bowl. Toss with the remaining 1 teaspoon extra-virgin olive oil and sprinkle with salt. Add reserved dressing. Toss to combine. Divide dressed greens and melon among plates. Garnish with pickled onion rings.