The arguments against magic are pretty weak, I do have to say. Making the statement that "magic is bad" because one is able to say "a wizard did it" is about as thoughtful as saying "mechanisms are bad" because one is able to say "a mechanic did it" when explaining how the floodgate is controlled by the lever on the opposite end of the map. After all, three mechanisms and you can hook anything up to a lever, so long as it's capable of moving - it's MAGIC!
In any case, one could simply define certain types of 'healing' magic - some don't have to even be conciously used.
Imagine, if you will, a dwarf who has the ability to unconsciously speed the healing of those around him. If we want to get technical, that could be his soul utilizing magic to speed up the healing process through an increased metabolism in both himself and others, with a result of faster aging for the wounded dwarves.
Alternately, a dwarven cleric of a god of Fire curing a patient of a disease could be invoking the deity to, at an atomic/molecular level, provide the energy necessary within all of the cells/structure of the virii/bacteria/fungi present in the dwarf's system, to disintegrate the organisms into their component atoms.
It's really not hard. Think of a natural step in the process of healing that you want to start from, and then instead of the antecedent occuring, simply say that "magic did it". The implication there would be that magic is necessarily more intelligent than dwarves and performs the dwarf's will (heal this dwarf of disease) even if the dwarf himself is unaware of how to accomplish his goal (kill the bacteria/virii/fungi).