The city of Rome at its height was about 1 million, and Rome's walls enclosed about 16 square miles. These numbers come from different sources, and it's not clear if the population figure refers to people inside the walls only. However, both are at least in the same ballpark as the figures given for Waterdeep.Well, consider - the population of England and Wales together, in 1500, was probably about 3 million. Is the city state/region of Waterdeep comparable to... half of Renaissance England? If yes, then fine. If not...
As others have observed, food is the real challenge. You can't feed a city that size by operating a few farms outside the walls; Rome was the capital of an empire, and it needed that empire. A whole fleet of ships was built to do nothing but ferry grain across the Mediterranean and stuff it into Rome's gaping maw. I don't claim to be an expert on the Realms, but I don't believe Waterdeep rules an empire at all. My impression is that it's a mercantile city with little territory of its own.