While it is true that many Prime Directive issues are seen when dealing with pre-warp cultures, that by no means makes it any less important when dealing with cultures that have achieved warp travel. The consequences for intentionally interfering with a society's natural order are just as potentially disasterous.
What if, after the doctor left the hospital ship, the Dinaali were forced to begin treating all patients with equal level of care? In a society that was already plagued with too much illness for its economy to support, a sudden shift in their treatment policies could cause their entire medical institution to collapse. This outcome is by no means a certainty, but it does demonstrate the unintentional results of interference.
As for the argument that the doctor is not bound by the Prime Directive because he's a piece of technology, I submit that would still make him the responsibility of Starfleet. It could be that Starfleet never imagined that a hologram confined to a sickbay could have the opportunity to influence a culture. If so, the fault lies with those officers that originally commisioned the EMH Mark 1.
My personal belief is that the doctor, much like Moriarty, had achieved a level of awareness that went beyond his original programming. That gives him rights, but it also burdens him with the responsibilities of other sentient beings. The doctor was considered to be in Starfleet; he had the authority to relieve the captain of duty if she was not medically fit in his opinion. He was given autonomy over his program, to expand his knowledge and his personality. In every respect, he was a person, a doctor, and a Starfleet officer. That makes him responsible for his actions, and bound by the Prime Directive.