Set aside a few raspberries for garnish. Place the remaining berries in a blender or food processor; cover and process until pureed. Add the honey, lemon juice and ginger; cover and process. Strain and discard seeds; set sauce aside.
Cut the cantaloupe and honeydew into three wedges; cut each wedge widthwise in half. Remove seeds and rind. With a knife, slice each piece of melon lengthwise toward narrow end without cutting completely to the end. Open into a fan shape.
On each dessert plate, place 2 tablespoons of raspberry sauce and a cantaloupe fan and honeydew fan. Garnish with reserved raspberries.