You might not even get a public defender. Is an expired license at the same level as a speeding ticket in your area? If so, your government might only assign public defenders to ‘criminal’ charges with potential jail times. Just be aware of that before you start desperately trying to find one that doesn’t ‘exist.’
The obligation to treat patients who are in need in an emergency setting IS a legal obligation in the US. If a patient is refused treatment at an emergency room, both the doctor and hospital can get gigantic fines. I don’t remember the max off hand, but it’s somewhere along the lines of $50,000 and $1.5 million, respectively. The law in question is EMTALA, or the emergency medical treatment and active labor act. A patient must receive stabilizing treatment, or be stabilized to the best of the hospital’s ability and transferred to appropriate care.