Achilles and Agamemnon have an explosive falling out over Briseis. The two had taken as war booty the beautiful Chryseis and Briseis. Agamemnon took Chyrseis as his possession and Achilles the other young woman, both of whom were dehumanized into war loot.
However, they were humans, and Chryseis's father offers a great deal of ransom for his daughter. When Agamemnon refuses it, Apollo sends a plague that can only be lifted if Agamemnon relinquishes Chryseis. He does so but feels it is his right to take Briseis as a replacement. Achilles is deeply offended. He goes off to his tent, sulks, and refuses to fight in the war against Troy, leading to grave losses for the Greeks.
Each of these men are at fault for being self-centered and insensitive. Agamemnon pulls rank in a blatant power play because he is solely focused on his own needs and sense of injury. He doesn't stop to think of the impact of alienating his most important soldier. Achilles is likewise self-centered. He dwells only on his own sense of injured pride rather than the bigger picture of the needs of the Greeks. He is willing to throw the Greek army under the proverbial bus (or chariot) because of a quarrel over war loot.
The argument does set off a chain of events in which Achilles makes the decision to choose the path of a short life of great glory rather than a long life of obscurity. The argument could have been avoided if either man had decided to act with greater wisdom.
It also was not necessary that this particular incident be the precipitating event for Achilles's decision. Given that Achilles is repeatedly hot-tempered, proud, and impulsive, it seems that if this quarrel had not arisen, something else would have upset him and triggered him to warfare. It is hard to imagine that someone as hot-headed as Achilles was not going to follow glory's route. In this case, I would argue that character, rather than a single event, determined destiny.