The thing that really caused the western empire to fall was the civil wars between Theodosius I and the usurpers Magnus Maximus and Eugenius. The western army was essentially decimated / dismembered after the battle of Frigidus. Gibbon's and the like blame Christianity and also excessive taxation for the fall of the west. A real good book "The Fall of Rome: And the End of Civilization" by Bryan Ward-Perkins demonstrates that the living standards for the Western Empire were very good right up to the fall of Rome in 410 and that in certain parts of the western empire, they did not fall that much for a while (50 to 150 years). Other parts (Britain) fell quickly.
Really it was the western army's disintegration that led to Germans crossing the a defenseless Rhine. Had Theodosius come to terms with one of the usurpers in the west, thing may have just rolled along for the west...
Indeed, the Eastern Roman Empire was Christian and lasted another thousand years.
Correlation does not prove causation, but anti-correlation does disprove causation. Rome got more moral, and fell. It’s pretty safe to say that personal immorality was not the cause of the fall.