One way to justify income inequality is the incentive argument. The gist is that income inequality is necessary as a motivating factor: if people could not get rich, then they would not have the incentive to work hard, innovate, invent and so on. The argument requires the assumption that hard work, innovation, inventing and so on are good; an assumption that has some plausibility.
This argument does have considerable appeal. In terms of psychology, it is reasonable to make the descriptive claim that people are motivated by the possibility of gain. This view was held by Thomas Hobbes and others on the grounds that it matches the observed behavior of many (but not all) people. If this view is correct, then achieving the goods of hard work, innovation, invention and so on would require income inequality.
There is the counter that some are motivated by factors other than achieving inequality in financial gain. Some are motivated by altruism, by a desire to improve, by curiosity, by the love of invention, by the desire to create things of beauty, to solve problems and other motives that do not depend on income. These sorts of motivations do suggest that income inequality is not necessary as a motivating factor, at least for some people.
Since this is a matter of fact, it can (in theory) be settled by empirical research. It worth noting that even if income inequality is necessary as a motivating factor, there remain other concerns, such as the question of how much income inequality is necessary and how much is morally acceptable.
Interestingly, the incentive argument is a two-edged sword: while it can be used to justify income inequality, it can also be used to argue against the sort of economic inequality that exists in the United States many other countries. The argument is as follows.
While worker productivity has increased significantly in the United States income for workers has not matched this productivity. This is a change from the past, the income of workers generally went up more proportionally to the increase in productivity. This explains, in part, why CEO (and upper management in general) salaries have seen a significant increase relative to worker income: the increased productivity of the workers generates more income for the upper management than it does for the workers.
If it is assumed that gain is necessary for motivation and that inequality is justified by the results (working harder, innovating, producing and so on), then the workers should receive a greater proportion of the returns on their productivity. After all, if high executive compensation is justified on the grounds of its motivational power, then the same principle would apply to workers. They, too, should receive compensation proportional to their productivity, innovation and so on. If they do not, then the incentive argument would entail that they would have no incentive to be as productive, etc.
It could be argued that top management earns its higher income by being responsible for the increase in worker productivity, that the increase in worker productivity is not because of the workers but because of leadership being motivated by the possibility of gain. If this is the case, then the disparity would be justified by the incentive argument: the workers are more productive because the CEO is motivated to make them more productive so she can have an even greater income.
However, if the increased productivity is due mainly to the workers, then this counters the incentive argument: if workers are more productive than before with less relative compensation, then there does not seem to be the alleged connection between incentive and productivity required by the incentive argument. If workers will increase productivity while receiving less compensation relative to their productivity, then the same should hold for the top executives. While there are other ways to warrant extreme income inequality, the incentive argument has a problem.
One possible response is to argue there are relevant differences between the executives and workers such that executives need the incentive provided by extreme inequality and workers are motivated sufficiently by other factors (like being able to buy food). It could also be contended that the workers are motivated by the extreme inequality as well, perhaps they would not be as productive if they did not have the (almost certainly false) belief that they will become rich.