If they receive Federal funds you are correct. And the majority (I don't know about vast but I don't have any numbers in front of me to argue the point) of hospitals do receive some sort of federal money. Under the Federal Emergency Medical Treatment and Labor Act private hospitals are required only to screen patients seeking emergency treatment and stabilize them for transport to a public hospital if an emergency exists. That's not the same as treating them. That said, most hopitals prefer treating the patients to risking the liability of transferring them to another facility.
Hospital Liability for Failing to Comply with the Federal Emergency Medical Treatment and Labor Act in Florida

In any case. emergency care is the most expensive and least efficient method of distributing healthcare.