I'm not sure that would have helped that much really. The Eastern Empire had been propping up the West for years with trade and money. By splitting off, the East was able to keep all that funding for itself. However, internal/external struggles, the Forth Crusade[1] and political intrigue ended up being the downfalls. I do think there is some "what if?" factor in all that. Such as what if the unified empire had not wasted all those resources on the civil wars.
In some ways, the Western Empire was sort of doomed from the time of Augustus onward when their embedded fear of the German tribes was established[0]. The defeat at Teutoburg Forest and the loss of around 20,000 Roman legionaries stopped all expansion into Germany and most likely lead the the eventual confrontation with the Goths. There's also a "what if" due to the Huns and pushing the Goths across the Danube, but if it were not the Huns, it could have been other factors as well (famines or another barbarian tribe).
The Eastern Empire almost reconquered the Western half in the 6th century under Justinian[2] and Belisarius[3]. However, internal bickering and mistrust led to squandering much of the gains that were won.
The Islamic forces were also able to capitalize on the chaos brought on by the perpetual wars between the Eastern Empire and the Sassanids[4] as well as the various barbarian tribes (such as the Bulgars and the various Gothic tribes that filled the vacuum of the former Western Empire) that still preyed on the Constantinople and other areas. There was just too many fires to contain them all at once and when one was put out, another was started in another region.
Honestly, I think it was for the best the Roman Empire ended for the sake of renewal (as much as I love Roman History). It had started to stagnate by the forth century and the more Greek side in the Eastern Empire was far from being the intellectual and cultural epicenter that it was during Classical Greek times. Sure, they built great things such as the Hippodrome and the Hagia Sofia, but those were just reincarnations of similar things done under the unified empire.
[0] http://en.wikipedia.org/wiki/Battle_of_the_Teutoburg_Forest
[1] http://en.wikipedia.org/wiki/Fourth_Crusade
[2] http://en.wikipedia.org/wiki/Belisarius
[3] http://en.wikipedia.org/wiki/Justinian
[4] http://en.wikipedia.org/wiki/Sassanid_Empire