People who have enjoyed a relatively high level of education tend to have higher wages. They are more likely to be employed. And their marketable skills give them a competitive advantage. It would seem to follow from this that countries with low percentages of the population having completed some specified level of education (say secondary education) should also be countries that have relatively high levels of income inequality. However, that’s not really the case (the correlation is obviously weak if it’s there at all).
It’s also true that educational attainment levels have risen in countries where income inequality has risen. All this would suggest that it’s not insufficient education that causes income inequality and that it’s futile to try to reduce income inequality by way of broadening the levels of education in a country.
However, that statement may go a bit too far. Education probably helps, but its effects are swamped by the effects of other factors that go the other way and aggravate income inequality. For instance, wage premiums aren’t the simple product of one’s education level. The type of education also matters (engineers will probably always earn more than philosophers), as do some noncognitive traits that are fostered by education but that are also more difficult if not impossible to equalize through education (such as discipline and intelligence). Add to this all the other elements that determine the levels of income in a society – the nature of one’s parents, peers and neighborhood, the social selection of desirable skills, international wage competition, regulation, corporate governance, taxation etc. – and it becomes clear that education alone can’t possibly produce a lot more income equality.
But that doesn’t mean it can’t help or that it shouldn’t be promoted for other reasons. Education is a good in itself, regardless of its effects on inequality. Furthermore, education promotes social mobility (the correlation between parents’ earnings and children’s earnings). See here; a much stronger correlation, especially without the US.