Public hospitals in most states can't refuse people who seek treatment in emergency rooms. Additional services beyond the emergency room can be refused by public hospitals. Private hospitals, on the other hand, can refuse people without insurance.
How to treat injured person when they present themselves without considering payments is difficult policy issue for local and state governments and the entire country. Unfortunately, a patchwork quilt of contracts, laws, regulations, tax laws, and government regulatory requirements, exists which doesn't always fairly balance these competing needs and interests.
The answer can also vary depending on the applicable state law, and the type of hospital institution. In some states, the state laws provide that public or non-profit charitable institutions that receive tax-exempt benefits or state or local funding are required to treat emergency cases, and in some instances other type of cases or admissions, unless they don't have the equipment or resources at that time to properly treat the person. Dependent on whether that hospital participates in certain contracts or other programs, the burdens to treat or continue treatment can be greater or differ.
Certain federal laws, including EMTALA, require emergency treatment of any person who presents at least until the patient is stabilized, unless they don't have the equipment or resources at that time to properly treat the person. Those same laws apply to public and private hospitals with emergency room treatment facilities.
Since, legal compulsion to provide a service free of charge is a rare circumstance, analyzing the rights of the patient, the hospital, the treating physicians, and the local and state taxpayers is a complex legal question. Often, a acute-care hospital or emergency room is not the right facility to render care for non-emergency medical treatment. More severely injured persons should be served there instead.
Maybe a local non-profit charity or clinic, or a local physicians group providing pro bono care can help instead.
Keep in mind that many communities have cost-free or reduced cost clinics and out-patient facilities that are intended to provide equal or better treatment at greater efficiency. Absent a medical emergency, you should explore whether those types of lower intensity medical attention are available to you before resort to the legal process.
Good luck in your pursuit of quality health care.
State Required Legal Ethics Disclosure: This Answer and any information contained in this answer is not intended to be treated as legal advice; And, this posting does not create an attorney-client relationship or privilege of any kind. This attorney licensed only in Georgia.