Without going into too much details, the "small bickering nations" stems simply from the fact that there were no lightning rails or stormships. The Roman Empire had a lot of trouble staying really united because of that -- if you had an emperor in Constantinople, he just wasn't going to be able to react to an usurpator seizing power in the West. So there were "vice-emperors" charged of administering a big chunk of the Empire, while the Emperor himself administered another.
It wasn't a very stable arrangement, so it fell; even though it made a surprisingly long time before doing so. But as long as it stood, there were periodically some general (or even a caesar) who decided he needed no longer pay tribute to the augustus and became an usurpator.
Another thing is that the Church discouraged mobility and travel, instead encouraging people to stay where they were. This was a good way of disuniting the Empire, as each province became more and more a nation, in a way.
Empires fragmented into kingdoms, and the European powers grew together, staying roughly balanced. The Antiquity became the Middle-Age which became the Renaissance which became the early Modern Era. Technology and the arts of war developed quite a lot throughout that time, until finally the biggest European powers decided it was futile to try to attack the neighbor that's just as well-armed as you, when there were so many other people who could be easily outgunned. And that's where whole continents were united -- by the Europeans. Guns trump bows.
The thing is, it wasn't just a question of better equipment. Even if the natives were outmatched in that respect, they still had a huge numerical advantage, be it in Africa, Asia, or the Americas. Europe could only afford to send a limited number of troops, after all -- especially considering the European powers were still wary of a neighbor who could jump on the opportunity to invade the mainland now that all the armies are oversea. So, why did the European win and manage to colonize about everybody else?
Because all the other continents were as united as Europe, that is to say, not at all. Tribe A made some deal with the invaders so that they could both attack Tribe B (Tribe A's hereditary enemies) and get rid of them. Then once that was done, the Europeans just allied with Tribe C in order to wipe out Tribe A. And so on.
Which just proves -- you give humanity far too much credit if you think that only the Europeans have really practiced internecine warfare!
