Place the water in a small saucepan and sprinkle with the gelatin; set aside.

Place the berries in a blender and blend until smooth, stopping the motor and scraping down the sides of the bowl with a rubber spatula as needed.

Heat the reserved gelatin mixture over low heat, stirring until smooth, about 1 minute. Add the berry purée, sugar, and salt and stir until the sugar has dissolved and the mixture is warm to the touch, about 5 minutes. Remove from the heat. Place a fine-mesh strainer over a medium bowl and strain the berry mixture, pressing down on the solids with a rubber spatula; discard the solids. Refrigerate until cold, at least 1 hour.

Add the milk and lemon juice to the chilled berry mixture and stir to combine. Freeze in an ice cream maker according to the manufacturers instructions. Transfer to a container with a tight fitting lid and place in the freezer until completely frozen, at least 3 hours. Let sit at room temperature for a few minutes before serving. The sherbet will keep for up to 1 month.