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.
What interests me as a Christian is how many Christians actually think the empire fell because of the faith embraced by the empire (almost a perverse pleasure in that thought) or how many think Rome got more decadent as the faith spread. Gibbons was anti-Christian and yet so many believers think he was right...