Judging from the experience of African Americans, is there a correlation between race, class and access to medical care in the USA?
Absolutely, and this is not unique to African-Americans, rather, it is much more closely tied to class and poverty. Over 50 million Americans are uninsured, and as the US has a for-profit health and insurance system in place, there is little incentive for private practitioners and hospitals to locate in poverty-stricken neighborhoods and regions. Because the African-American population is concentrated in southern states (which tend to be poorer) and the inner cities (same issues), they are disproportionately struggling with lack of access to health care, and other chronic medical issues.
It is difficult, if not impossible, to have a health system that is motivated by the usually exclusive goals of public health and profit. Successful systems tend to isolate themselves from the economic pressures of competition with government subsidies and regulations, while a profit-based system will, by design, exclude the poorest citizens of a society.