On the subject of EMP(Medical), it is has become a common linkage in games since at least 2000 with D&D3e. That said, Empathy is the "interact with people" stat. Diagnosing a medical problem and assessing the treatment applied has a fundamental component involving this. Even House has a decent EMP, he is just too surly to use his likewise decent Persuasion for charm.
That just doesn't wash ... at least, not for the commonly accepted definition of "empathy".
Some of the people absolutely best at interacting with others, with the most charm and the highest ability to get people to do precisely what they want them to do — e.g. pathological narcissists and psychopaths — have by definition zero empathy.
And connecting to Medical — well, there won't be much interpersonal interaction with an unconscious or comatose person, something a medic can be expected to do with some regularity, and still be efficiently able to determine what's wrong with them, and what to do about it.
In fact, determining whether the person is just that — comatose or merely unconscious — is in itself a vitally important diagnostic task, but requires exactly zero ability from the medic to understand and share the patient's emotional experience
, which is what "empathy" is
Connecting a Medic skill to an Empathy stat having become a trope in RPG's doesn't by necessity make it a good idea.
But instead of trying to re-write the dictionary definition of "empathy", perhaps it would be better to consider calling the stat something else? "HUMINT"...?