Halve the melon and remove the seeds. Scoop out the flesh using a melon baller. Add the melon balls and the geranium stalks to a punch bowl.
Mix the Campari and 2 tablespoons of sugar with half of the Prosecco. Add the Campari and Prosecco mixture to the bowl. Place in a refrigerator and chill for at least 2 hours. Before serving, mix in the remaining Prosecco. Pour into glasses, garnish with geranium stems if you'd like, and serve.