In a food processor, add the tomatoes, basil, olives, and olive oil and process until finely chopped.
Season both sides of the scallops with salt and pepper. Rub each scallop with the tomato mixture. Fold each slice of prosciutto in half lengthwise, then wrap each scallop in 1 slice of prosciutto. Place wrapped scallops in a buttered baking dish, seam side down. Bake until scallop is cooked through, about 15 minutes.
In a medium bowl, toss the arugula with the balsamic vinegar. Season the arugula with salt and pepper. Place the arugula on a serving platter or divide among individual dishes. Top with the scallops and serve immediately.