Remove peel and pith from each grapefruit. Holding over a bowl, carve out flesh between the membranes, letting segments drop into the bowl (you should have about 2 cups segments). Squeeze membranes over bowl. Strain juice into a measuring cup. Transfer 1/2 cup juice to a saucepan; discard remaining juice.
Make the dressing: Add honey, lemon juice, and ginger to pan with juice. Bring mixture to a simmer, stirring occasionally. Remove from heat; let cool completely. Discard ginger. Gradually whisk in 1 tablespoon oil until emulsified.
Lay scallops, cut side up, on a rimmed baking sheet; season tops with 1/8 teaspoon salt. Heat remaining 2 tablespoons oil in a medium skillet over medium-high heat. Add scallops, seasoned side down; cook until edges are browned, about 4 minutes. Flip, and cook 30 seconds more. Set aside.
Toss arugula sprouts with 1 tablespoon dressing in a small bowl; set aside.
To serve, divide grapefruit segments among 4 serving plates. Top each with sprouts and seared scallops, dividing evenly. Drizzle each plate with 1 tablespoon dressing, and season with remaining 1/8 teaspoon salt and the pepper.