Prepare an ice-water bath; set aside. Bring a medium saucepan of water to a boil. Add basil; cook until basil turns bright green, about 30 seconds. Using a slotted spoon, transfer to ice-water bath until cool, about 1 minute. Drain; pat dry.
Stir together sugar and 1 cup water in a medium saucepan. Bring to a boil over medium-high heat, stirring constantly, until sugar has dissolved. Add basil. Reduce heat; simmer 5 minutes. Remove pan from heat; let cool completely.
Puree the basil mixture in a blender, about 1 minute. Transfer to a large bowl. Stir in honeydew juice. Refrigerate, covered, 8 to 12 hours.
Using the back of a spoon, press basil mixture through a fine sieve into a nonreactive bowl. Stir in lemon juice and salt. Transfer to an 8-inch square nonreactive baking dish, and place in freezer. Rake with a fork every 30 minutes until mixture is completely frozen, about 3 hours.
Arrange peaches on 6 chilled serving plates, dividing evenly; top with granita. Garnish with basil sprigs and flowers.