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 puree 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 a fork to scrape the crystals toward the center of the pan, then continue freezing, scraping every 30 minutes, until frozen, about 4 hours.