We can both say what we think both parties should have done, but this is entirely subjective and would come down to ideological differences and feelings.
> Where United went wrong
This is where we can find some truth to the matter. What does wrong mean here.
Each side could have averted this event. And each side lost something from this event.
Dr - Lost his dignity, became a celebrity with his face plastered around the world (good and bad - i.e. some people like myself think he is an idiot), physical injury, mental anguish, etc.
United - Share price, reputation, had to apologise, may lose customers, etc.
For both sides there are lots of negatives. United clearly had more to lose (tangible shareholder value and reputation) so they should have not created the situation.
I wish more people would look at this for what it is, which is childish behaviour from a professional adult, and just move on, but alas in today's world it seems that every event is being filmed and ready to go viral, so perhaps the next video we see will be me sitting on a plane that is being delayed hours while someone is being coddled and sweet talked to get off a plane they are being bumped from. This Dr may miss his appointments with his patients, but what about all the other hundred passengers who will be late or miss their connections.