1. Process melon, apple juice, and lime juice in a blender until smooth. Pour the mixture into a 9"x13" pan.
2. Place the pan on a level surface in your freezer. Freeze, stirring and scraping with a fork every 30 minutes, until the granita is firm but not frozen solid (3-4 hours).
3. Remove from the freezer and break up into small slivers. Pack into an airtight plastic container and freeze until solid (1 hour).
4. Remove from the freezer about 20 minutes before serving to soften slightly. Use a wide spoon to scrape the granita into shallow bowls. Sprinkle blueberries and raspberries over each portion and garnish with mint leaves.