Swordfish with Watermelon and Lime-Ginger Citronette

fresh lime juice (from 2 to 3 limes)

sugar

finely grated peeled fresh ginger

safflower oil, plus more for brushing

Coarse salt and freshly ground pepper

skinless swordfish steak, cut into 1 1/2-inch pieces

large scallions, white and pale-green parts separated, dark-green parts thinly sliced

head Bibb lettuce (about 8 ounces), leaves separated

small seedless watermelon, rind removed, cut into 1/2-inch-thick wedges

medium carrot, peeled into ribbons and julienned

Flaky sea salt, such as Maldon, for serving