You forgot the Eastern Roman empire crownd by the reign of Constantine. Rome didn’t fall in 44 bc, contrary to popular belief. In fact while Rome itself fell, the Empire simply moved east and lasted another 700 years or so.
So either way, the 'Empire' didn't last 1300 years.
And to answer your original question. I would say that the Roman Republic fell when it's people and leaders were no longer moral.
Many things that the Roman Republic started are still with us, though, and considered moral behavior.
/johnny