Profit from health care is not unethical. It's profit from health insurance that I find questionable. As I said, it seems like a serious conflict of interest. The insurer receives compensation in exchange for assuming risk. If the insurer is looking for profit, they are then motivated to optimize risk out of their system as much as possible. This can be done ethically (investing in the health of policy holders) or unethically (denying benefits whenever possible and/or leaving all but the lowest-risk clients out in the cold). History seems to indicate that private firms will take the latter approach when permitted to do so.
Health care is essential to the extent that pretty much everyone will need it in some form at some point. Also, as others have mentioned, emergency rooms can't really refuse patients. It doesn't work like conventional goods and services because an individual's needs are not necessarily predictable, and they are susceptible to catastrophic misfortune that could ruin them financially. The same concept applies to car/home/life insurance, fire protection, unemployment, and other risk-based "goods".
All insurance has issues. It's not good for controlling costs. For health care, the specifics about what should be covered are of course up for debate, and there are major questions about personal responsibility for one's lifestyle and the impact it has on health and healthcare needs. However, the private insurance "solution" doesn't seem to address these issues any more than socialized insurance would. Either we collectively cover the "problematic" individuals, or we collectively pay for their eventual ER visits because those same individuals didn't have preventative health care.