Combine melon, apple juice and lime juice in a blender; puree until smooth. Pour the mixture into a 9-by-13-inch glass or metal pan.
Place the pan on a level surface in your freezer. Freeze, stirring and scraping with a fork every 30 minutes, moving the frozen edges in toward the slushy center and crushing any lumps, until the granita is firm but not frozen solid, 3 to 4 hours.
Remove from the freezer; use a metal spatula or large spoon to break up the frozen ice into small slivers. Pack into an airtight plastic container and freeze for at least 1 hour more.
Remove from the freezer about 20 minutes before serving to soften slightly. Use a wide spoon or ice cream scoop to scrape the granita into shallow bowls. Sprinkle blueberries and raspberries over each portion and garnish with mint leaves, if desired.