1. Combine water and sugar in a small saucepan. Cook, stirring, over high heat until the sugar is dissolved. Transfer to a glass measuring cup and let cool slightly.
2. Puree watermelon in a food processor or blender, in 2 batches, pulsing until smooth. Transfer to a large bowl. Whisk in the cooled sugar syrup, yogurt, and lime juice until combined. Pour the mixture through a fine-mesh sieve into another large bowl, whisking to release all juice. Discard pulp. Pour the extracted juices into a shallow metal pan and freeze until solid, about 6 hours or overnight. Remove from freezer to defrost slightly, 5 minutes. Break into small chunks and process in a food processor, in batches, until smooth and creamy. Serve immediately or transfer to a storage container and freeze for up to 2 hours.

Make Ahead Tip: If frozen longer than 2 hours, break into chunks and puree in a food processor until smooth before serving.