Combine the raspberries, sugar and 1/4 cup water in a small saucepan and bring to a simmer over medium heat. Cook, stirring occasionally, until the raspberries have completely broken down, about 6 minutes.

Press the raspberry mixture through a fine mesh sieve into a blender;let cool completely. Add the watermelon and lime juice and purée until smooth. Pour the mixture into an 8-inch square stainless steel or glass baking dish.

Freeze until ice crystals begin forming around the edges, about 45 minutes. Use fork to scrape the crystals toward the center of the pan, then continue freezing, scraping every 30 minutes, until frozen, about 4 hours.