Ugh. I hate getting law nerdy, and not using meme's but your understanding is so beyond reality that I can't let it go in case someone believes you. In the United States, barring a special relationship or statute, no one has a duty to rescue, because a moral obligation does not equal a legal duty. People v. Beardsley, 150 Mich. 206, 113 N.W. 1128, 13 L.R.A., N.S., 1020. Only 10 states have some form of duty to rescue: California, Florida, Hawaii, Massachusetts, Minnesota, Ohio, Rhode Island, Vermont, Washington, and Wisconsin
As to your specific scenario, the Dr. has no personal responsibility towards the patient in the ER unless the person was already a patient, which constitutes that special relationship mentioned previously. It's easier to see when we change the location. The Dr. is at a restaurant having dinner and the person at the table next to him falls down with a heart attack. The Dr. has no obligation and is free to continue eating his dinner.
However, in your scenario, the patient walks into the ER. In this instance, the hospital is forced by the Emergency Medical and Treatment Labor Act (EMTLA) of 1986 to provide services to that person. Without any admission of the appropriateness of the EMTLA on my part, under the Dr's employment contract with the hospital, he is now obligated to provide services. He will, however, be paid for those services under which he is contracted.
So just to sum up, a) no one has a duty to rescue in the US baring a special relationship or statute, and b) the only reason your Dr. has to treat the man in the ER is because he has a prior contract to do so, for which he is compensated, therefore no slavery exists on the part of the Dr. However, the payment of the bill doesn't magically disappear. Either the patient still has to pay, or the government steps in and pays. In the case of the government paying, the government has confiscatoraly taken money from other citizens and used that money to pay the medical bills. And as you know, slavery is the confiscatory taking of one person's labor for the benefit of another.
Now for the legal disclaimer: The above is the opinion of the writer and does not constitute any form of legal advice because the writer has been out of the legal industry for more than a decade because being a lawyer sucks ... in his opinion. If, for some reason you're dumb enough to take a couple paragraphs from a meme website as justification to do something stupid, that's on you.