Place cantaloupe and honeydew melon in a large resealable plastic bag set in a shallow dish. For marinade, in a small bowl stir together lime peel, lime juice, and ginger. Pour marinade over melon in bag; seal bag. Turn to coat melon. Marinate at room temperature for 30 to 60 minutes, turning bag once or twice.
Drain melon, reserving marinade. On four long metal or wooden* skewers, alternately thread cantaloupe and honeydew melon.
For a charcoal grill, grill skewers on the rack of an uncovered grill directly over medium coals for 6 to 8 minutes or until melon is warm and shows grill marks, turning occasionally. (For a gas grill, preheat grill. Reduce heat to medium. Place skewers on grill rack over heat. Cover and grill as directed.)
In a small bowl whisk together the reserved marinade and the honey. Drizzle over warm kabobs.
If using wooden skewers, before grilling, soak them in enough water to cover for 30 minutes. Drain before using.