As far as I know, there are no information that I've come across in reliable books or peer-reviewed journal articles that indicate slavery had any significant role in causing the fall of the Roman Empire. Rather, I've come across incidental information in several works that suggest slavery helped the economy of the Roman Empire thrive during its various peaks. One example is how the Roman economy literally exploded with surplus after Trajan's conquest of Dacia and the subsequent mass depopulation of the region – the dude were rounded up and shipped off to Italy as war booty.