To make dressing: Whisk together lime juice, paprika, salt, and pepper in a small bowl and add oil in a slow stream, whisking until well blended.
To make salad: Toss cantaloupe and honeydew with half of dressing in a bowl. Decoratively arrange the arugula on a platter and then add the melon. Season with salt and pepper, to taste an drizzle with remaining dressing. Serve immediately.
Cook's note: It is important that the ham not be sliced as thinly as prosciutto usually is. Slightly thicker ham is easier to cut into strips and will stay separated in a salad instead of clumping together.