That is exceedingly hard to determine in the perpetual soap opera that were the Roman monarchy. It would be easier to say that the fall of the western empire was due to multiculturalism than to moral decay.
It could even be asserted that Christianity was responsible, Constantine I taking over just a little over a hundred years before it fell.
It would be easy to make an educated guess because you can look at the written record, consider how much it influenced court society and society at large and why.
Also, you’d have to account for how Eunuchs were employed before during and after the Empire’s resurgence in the Sixth, Eighth and Thirteenth Centuries.
Of course, multiculturalism is accompanied by moral decay, especially the soft Marxism of the Frankfurt School. http://en.wikipedia.org/wiki/Frankfurt_School
Of course, I suspect that despite the fact that the Eastern Empire was a multi-ethnic society, it was still able to forge a critical unity based upon the moral influence of the Orthodox Church which perpetuated the Classical heritage of the ancients and kept people’s mores in check to bring stability to the society, accounting for its tremendous resiliency.
It’s interesting in the tape of Nixon that he starts off by denying that his concerns are moral, but the more he talks about it, the more he realizes that his concerns are indeed moral ones. He’s worried about the children who will watch the cultural marxism of Karl Reiner on All in the Family, a supposedly family show showing indecent filth like that to children... in fact, I think I was 12 myself when I watched the episode that Nixon is talking about.
Are you familiar with Nixon’s controversial tapes dealing with this issue? They’re available on youtube.
http://www.youtube.com/watch?v=TivVcfSBVSM
Given that the Eastern empire survived for another 1000 years, that’s not a very strong argument.