When it was agreed that the umpires screwed up the famous George Brett pine tar incident the rest of the game was replayed from the point of Brett's home run. Maybe they could do that. Yeah right.
The rules of Major League Baseball allow a manager to inform an umpire that the game is "under protest" in certain cases, which means the president of the league in question (the American League) will go through a formal review of the game and make a decision as to whether or not the protest will be upheld. The incident you cited was one of the rare cases where a protest was upheld.
It is important to remember that a manager cannot lodge a formal protest over an umpire's judgement. Formal protests are only permitted when a manager is questioning an umpire's application of the rules. For example: if a batter bounces a long fly ball over the wall and the umpire allows him to run to third instead of awarding him a ground-rule double, the opposing manager's formal protest will likely be upheld and the league president will have to make a decision as to how much of an impact the umpire's mistake had on the outcome of the game.
Because of this limitation on formal protests, and because of the fact that major league umpires are generally very competent when it comes to applying the rules of baseball, formal protests are usually only filed in response to very rare or even unprecedented game situations (two balls in play at one time, fans interfering with balls in play, baserunners getting hit by batted or thrown balls, etc.) or incorrect applications of rules that are easy for an umpire to overlook (a team batting out of order, for example).