if you take the city of Roma, then no -- gays were more rampant a few centuries earlier -- read about Elagabulus the Roman emperor in 218 AD who had a blonde slace named Hierocles (Gallic probably from modern day Turkey) whom he called his husband, He was described as having been "delighted to be called the mistress, the wife, the Queen of Hierocles" and was said to have offered vast sums of money to the physician who could equip him with female genitalia (Cassius)
This guy did lead to the decline before Domitian, but this was in 218-222 AD, 200+ years before the city fell to Roman-trained, Alaric in 410 and then Odoacer, a Roman cultured Germanic defeated Romulus Augustus in 476 AD. This was not due to homosexuality.
The Romaioi had the same problems that the Empire in the West had, although Orthodoxy in the East as in the West served as a stabilizing factor, but as in the Latin West, the Orthodox East had problems maintaining any kind of stable source of manpower loyal to the State. Part of the problem was too much centralization and high taxes, debasing the coinage, but all of that is ancillary to the moral, and there are many accounts of the immorality of the Romaik Empire told even by Monks and Bishops, threatening of a deluge. Indeed, when the city fell in 1453, most of the troops manning the walls were Italian and German mercenaries.