First of all, at present the income gap is so extreme as to make it impossible for some to have meaningful access to food, shelter, and clean water. As long as people could do better if they could pitch a tent in a national campground and just forage and hunt, society has failed utterly. No society, no social contract, can be justified as long as that degree of inequality is the case.
Second, government, even in the most minimalistic of markets, will always intervene to protect the rich. A military will do so, for example, as will police officers and fire departments. The rich get a lot out of society, while the poor have much less to lose from society being less stable or safe.
At present, virtually every non-failed state on the planet has a national bank. Most successful economies have had subsidies to private industry, technology funded by the public sector that benefited the private sector, etc. As long as government is acting to do anything in the economy, from affecting monetary policy to education, it should do so with an eye toward reducing the income gap, all else held equal.