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.
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 an ice cream maker and freeze according to manufacturer's directions. (Alternatively, pour 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.
Melon selection & storage:Look for symmetrical unblemished melons, without flat sides, that have a creamy yellow spot on the bottom indicating ripeness. At 92% water, this fruit should feel heavy when you heft it. Precut melon flesh should be dense, firm and appear moist.Store in the refrigerator for up to a week or keep in a cool, dark spot. Cover the cut surface of melon with plastic wrap and refrigerate.