Balkanizing Rome is impossible due to the different doctrines of public law.
China was always made up of different kingdoms, only loosely controlled by the so-called emperor. The kings were nothing more than nominal vassals of the emperor, often fighting against each other or even against the empire. Within their kingdom, the kings were absolute monarchs, checked only by tradition and their advisors.
Rome however is a republic, even if it was ruled by a emperor until recently. Even if the emperor ruled without senatorial or popular control, the emperor was legally nothing more than a simple magistrate of the republic. The Roman Emperor couldn't "divide" his empire among his sons (since it wasn't his private property), he could only appoint his sons emperors of different parts of the united empire. Even if there were, for military or administrative purposes, multiple emperors, there was always only one Roman people and one Roman Republic.
Another reason is geography. The core of the Roman Empire is the Mediterranean Sea and its coasts. Both capitals of the Empire, Rome and Constantinople, as well as the most important cities like Carthage, Antioch, Alexandria or Massilia are located on the sea. If it wasn't for the Mediterranean, which facilitated trade and troops movements, Rome may well have fallen to the opposing interests of its provinces. Never forget how close Rome came to balkanization when during the industrialization its economic focus shifted towards the north! Discussions about secession from the "backward agricultural south" were common in Colonia, Treverorum, Londinium or Eburacum as late as 1950!
China however has no such uniting sea. It has rivers, that's right, but they're linking only the west with the east, not the north with south. Therefore, they are often borders between northern and southern kingdoms, and the cultural differences between northern and southern China are very distinctive.
Add to that diplomatic reasons (Japan, Mongolia and India always schemed to prevent a Chinese reuinfication, while no European powers could ever have balkanized Rome) and cultural reasons (the different languages of China compared to the different dialects of Latin) and you understand that the balkanization of China was multicausal, but somewhat inevitable.