Most of the associations discussed are actually between income level and health, which is quite different from inequality per se. And the research is pretty clear that having a low income and being poor in general is terrible for one's health. But the story about how inequality affects health outcomes is far more complex - and interesting.
Income inequality can affect one's health in several ways, all involving the causal influence of inequality on other economic factors that in turn directly affect health. An example might be income itself: higher inequality might cause a larger share of the population to draw in low incomes in the absolute sense. Additionally, inequality might affect the political structure: the smaller the group of high-earners, the more easily political collaboration becomes, setting up a vicious cycle of institutional development that enables rent-seeking and economic sclerosis. A similar story presents itself for culture and social institutions.
The closest inequality gets to directly affecting health outcomes is through the pathway of psychology. It's possible that the greater inequality becomes, the more psychologically damaging being at the bottom of the income distribution is. High levels of stress, regret, feelings of inadequacy and depression are correlated to worse health outcomes. This seems especially plausible due to the high visibility of media displaying the lifestyles of top-achievers.