Prepare the Melons: Using a melon baller, scoop out balls from the honeydew, cranshaw, and watermelon.
Place all the melon balls into a large, deep bowl.
Infuse with Vermouth: Pour the sweet vermouth over the melon. Gently toss to combine, using large spoons to lift the fruit from the bottom and ensure everything is lightly coated.
Chill and Marinate: Cover the bowl tightly and refrigerate for at least 2 hours. During the chilling time, gently stir the mixture two or three times to allow the flavors to meld.
Serve Beautifully: To serve, transfer the chilled, vermouth-soaked melon balls to an elegant glass serving bowl or individual cocktail glasses.