I think it all depends on what he means by "inequality". Inequality of result arising on a level playing field is one thing - that is what progs & dems rail against - but inequality arising because there are different sets of rules for different classes of people - typically, one set for the rich and another, less favorable, for the poor - is at the root of social evil.
The Pope is (usually) more concerned with the less well-off parts of the world, the countries and societies where there are different sets of rules for the rich and the poor, or for the white and the black (not really anymore, except in affirmative action USA), or for Tutsi and Hutu, or .... Of course, not only is that state of affairs the result of an undeveloped society, it is also the result of imposing socialism/fascism on a country, whether developed or not. That is already happening here with Obastardcare because, as the WSJ pointed out yesterday, a two-tiered health care system is already emerging in the US, with the rich getting one type of health care - much better health care - and the rest of us being left with the dregs. This sort of inequality is the result of depriving a society of equality of opportunity.
Inequality of opportunity is most definitely a major cause of social evils, and the best cure humanity has yet found is a culture premised on free markets, robust private property and contract law effectively and objectively enforced, and government primarily involved in making sure that the predicates necessary for these to work, e.g., anti-trust rules and sensible rules that lessen the information disparity between various classes of market participants. In other words, the US and, more generally, free-market capitalism.