In a blender, process watermelon until smooth. Strain through a fine-mesh sieve into a bowl, pressing with a spatula until you have 3 cups of juice. Discard solids. Add sugar, corn syrup, lemon juice, and salt to watermelon juice, whisking until sugar is dissolved. Whisk in half-and-half. Refrigerate until very cold, about 3 hours.

Pour mixture into an ice cream freezer; process according to manufacturer’s instructions. Let stand at room temperature until slightly softened before serving. Scoop into dishes. Garnish with watermelon and lemon slices, if desired.