Huh. That's certainly interesting.
OK, let me make (part of) my point explicit: Inequality is frequently posed as a causal factor for a lot of things in ways that make no sense or when something else is actually meant. Such as:
1. Using inequality in place of poverty. The paper you link to explicitly excludes the simpler poverty hypothesis, but most argument I encounter on the internet doesn't bother to. (And then there's the tendency to jump to inequality as primary explanation when there's clearly something else going on screwing things up...)
2. In some cases inequality is posed as a causal factor in cases where the relation is clearly logical or compositional, not causal. People make statements basically analogous to "all-cause mortality causes murder". You could say the causation is backwards there, but in truth there's no causation at all -- it's a logical relation, not a causal one.
3. To use what's discussed in this paper as an example, sickness, like poverty, is fundamentally local; if you look at it at the group level you're simply aggregating what's going on with individuals. By contrast inequality only makes sense at the group level; it's fundamentally nonlocal, in a sense. The authors of this paper you link address this, noting that they're not committing an ecological fallacy because they're only looking at health outcomes at the group level. Nonetheless health outcomes at the group level are fundamentally an aggregate of individual health outcomes, so the question remains how individual health outcomes are affected. The authors provide some plausible speculation on this point; on the whole I think they've supported their point pretty well. But I think in general we should be quite skeptical of such explanations without a good reason to the contrary, because causation is fundamentally local; if you need to look at a property of the whole group when discussing individual outcomes, that's a point where you should stop and ask if that's really what's going on here, or whether there's a simpler explanation.
Worth noting is that the paper you link to, though it mostly discusses this nonlocal measure of inequality, when it comes time to talk about a mechanism, about "biological plausability", they talk about local gradients in social status. And if that's the real explanation then I think talking about this in terms of "income inequality" is misleading. (The authors get around this, as I noted above, by saying they're only going to focus on the group outcome, not individual outcomes. But for the reason I said above, to my mind that's just evading the problem, or solving an easier problem in place of the real one.) Because then it's not nonlocal inequality that's causal, but rather local inequality, and not inequality of income but of social status. Which is not the thing the authors are discussing in most of the paper! The thing the authors are claiming is causal, is not the same thing they actually posit as a mechanism. One needs to be careful here. These details matter.