M. Tirith is not a melting point of two great cultures as Byzantium was of Asia and Europe.
But yeah, nice analogy. Minas Anor, then Minas Tirith; Byzantium, then after Constantine died it was renamed Constantinople (although I'm not very sure if Wiki is right). MTirith became MTirith because it serves as a guard against Minas Morgul, and at some point defended by the Great River. Although I don't think MTirith has much gardens before Aragorn; the only gardens were at the Houses of Healing, right? Constantinople; a city where an emperor could sit, readily defended, with easy access to the Danube or the Euphrates frontiers, his court supplied from the rich gardens and sophisticated workshops of Roman Asia, his treasuries filled by the wealthiest provinces of the empire, as Wiki says.