For the simple syrup: Combine the sugar and 1 cup water in a small saucepot over medium heat. Bring to a simmer and cook, whisking occasionally, until the sugar dissolves, about 5 minutes. Remove from the heat and let cool to room temperature. Refrigerate in an airtight container for up to 2 weeks.
For the gimlet: Add the watermelon to a cocktail shaker and muddle. Fill the shaker with the ice. Pour in the gin or vodka, lime juice and 3/4 ounce of the simple syrup (about 1 1/2 tablespoons). Shake well and strain into a chilled martini glass. Garnish with the watermelon slice.