Use a knife to slice a small section off the bottom of the watermelon so that it will stand on its end. Slice ½-inch off the top then use a spoon to hollow out the watermelon.
Reserve the meat and juices from the watermelon in a large bowl. Once the watermelon is completely hollowed, set it aside and add the juice and meat to a blender.
Blend until pureed. Use a fine mesh strainer with cheesecloth set in it to strain the pulp from the juice back into a large mixing bowl. The watermelon should yield 3 cups of juice.
Combine the 3 cups of watermelon juice with the gin, lime juice and basil, stirring to combine. Pour the cocktail mixture back into the watermelon with ice.
To serve, ladle into ice-filled tumblers and garnish with lime wedges.