Bring water to a boil in a small saucepan; remove from heat and add sugar. Stir until completely dissolved. Pour syrup mixture into a heat-proof container, cover halfway with plastic wrap, and place in the refrigerator until completely chilled, 45 to 60 minutes.
Put blackberries, strawberries, and peach slices into a blender and blend until pureed. Add the fruit puree to the chilled sugar syrup and strain the mixture to remove the seeds. Stir in lemon juice. Cover and refrigerate until the mixture is chilled, about 1 hour.
Pour mixture into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.