Place the melon in the bowl of a food processor and process until smooth. Add the lemon juice, vodka, and sugar and process for another 30 seconds. Place the mixture into the refrigerator until the mixture reaches 40 degrees F; depending on the temperature of your ingredients and refrigerator, this could take 30 minutes to 1 hour.
Pour the chilled mixture into the bowl of an ice cream maker and process according to manufacturer's directions. Transfer the sorbet to an airtight container and place in the freezer for 3 to 4 hours before serving.