Hello kruly:
Actually health administration does fall under the umbrella of public health. There are different aspects of public health: biostatistics, behavioral health, epidemiology, health policy & management. There are people who have an MPH degree and are hospital CEOs. There are also programs that offer the MPH instead of the MHA and prepare students for health administration. For example, Yale, UC Berkeley, and UCLA offer MPH programs instead of MHA programs and students from those 3 institutions get awared hospial administrative fellowships all over the country.