Actually, I tend to agree with the poster of the question, that christianity actually played a prominent role in the downfall of the Empire.
Montheistic religions were actually quite old, much older than the empire. But because of their very nature they had trouble spreading. This is because the 'one-god' is often exclusively for one kind of tribe or people. Such as with judeism, for the Jews. The remarkable thing is that christianity became a mono-theisme for all. Of course the Jews are still Gods people, but Jesus became a bridge to all none Jews. (actually, there appear to have been two streams, after Jesus was crucified. One stream believed in keeping christianity for the Jews, the other, smaller group, that is was for all. Those of you who know the new testament will remember the Roman soldier thus converted)
Anyway, what signifies mono-theistic systems, like judeism, is the fact that they tend to form close nit, nearly or completely independent, communities, that awnser only to God. (like monastic communities)
Something else important is that the one-God does not bear other Gods. Mono-theisme is not a tolerant system, it has an important element of 'truth', that defies all others, and has a great drive to evangelicise, if need be by force. Holy wars, like crucades and jihads, are the proof of this, but also the marriage between religion and state.
And I believe that these two things meant, for a great deal, the downfall of the organization of the empire. A testament to this is that christian states never had the capability to grow as large as the empire, because the new mono-theisme made it harder to unite in a state. It took us many many centuries to seperate church from state.
The Roman Empire was a city state system, but with many monastic communities springing up, this system lost its appeal. Communities downsized in fact, and became virtually independent.
A law that was passed that actually liberated the tax system, more or less privatising it to large land-owners, meant that the legions no longer were supportable. And with the loss of the legions, Rome lost one of the greatest speaders of culture and control. Legions became allied to local lords.
And why did the Byzantine empire, as eastern part, stick it out longer? In fact, both Rome and the Byzantines held some sort of control. Rome was not 'gone'. The more than 1000 year old institution of the Pope of Roman Catholism sprang up, and had a great deal of influence in northern europe. Rome essentially chose to continue influence through being representitive of God himself, as the Roman emperors who had declared themselves God.
The Byzantines kept more control, because they held on to the military ways, while they in fact parted from the way of Rome by religion, as orthodox christians (which later gave them conflict with the west and Rome). But the Byzantine empire broke up later, also because of mono-theism. They were torn up between the Roman Catholics and the Moslim states, that, later than christianity, also became a one-God for all system thanks to Mohammed.
Personally, I would have preferred the old religions, that usually gave everybody the freedom, whoever your king or emperor, to excersise the religion and life style you wished.