"The City of Splendors laid claim to the surrounding lands for approximately 100 miles (160 kilometers) beyond its walls." (p.299, FR campaign guide, 3E).Waterdeep is not the capital of anything. There’s no empire nor is it even really master of any territory.
Waterdeep is an Oligarchical City-state; it rules itself and the territory it controls (100 miles beyond its walls). It controls a massive seaport that also doubles as a Spelljammer port, making it both a sea and space trading hub.