Can gender inequality contribute to the spread of HIV?

The two issues are more or less independent. Poverty contributes to the spread of HIV since protection is relatively more expensive and since poor people, on average, have shorter lives, they care less about a disease which may take decades to manifest. However, poverty is very different from income inequality. (You can a high level of wealth inequality, but have very few people who are at the poor end, and conversely, you can have almost universal poverty.)