Why there is so little healing magic in Dresden Files?

Magic is complicated and difficult. The human body is complicated and difficult.

Put them together, and becoming a true master is like getting PhD doctorates in two vastly different fields. While there may be many little magics that tread in healing waters, like Elaine's handling of minor injuries, healing serious or mortal injuries with magic would take not only a surgeon's skill to know what to effect and when, but also a master wizard's skill to put those energies to work in exactly the right amount at exactly the right time.

Building things has always been harder than blowing them up. Magic still has to work in the real world with real physics (at least in The Dresden Files), or in this case in real body parts inside real bodies. It's not as simple as sealing skin; diagnosing injury is an entire field to itself, let alone the treatment. Telling a body to just "grow back" without proper guidance and knowledge behind it is probably a good way to create a cancer spell, presuming a prompt outgrowth of tissue just doesn't kill the patient.

It's also treading grey areas that tend to invoke grey cloaks. Working spells on other people's flesh can invoke Second Law warnings (thou shalt not transform others). It doesn't matter if you didn't mean it; if you didn't do it right, you fail. If you screw up badly enough, bam, you've killed someone with magic, First Law violation equals swift and certain execution. So the Council is really the only resource you could study under safely (presuming you can find a master who knows the field), and they aren't the most open bunch.

Injun Joe is probably the foremost healer mentioned in the series, and it's mentioned that he goes back to modern university every now and then just to keep his medical skills sharp.

I pondered that the Wardens might know some magic in this vein, but given their penchant for destructive magic and that they need to be efficient, modern military field medicine techniques probably serve them far better.

It not so much that there isn't a lot of healing magic in the DF, it's that it mostly happens off screen. Harry is basically a magical thug that has great strength and a serious proclivity for the kaboom type of magic. As he progresses through the series he does improve and gain some subtlety in his power and build on his other magical skills: thaumaturgy,potions, artifice. But since Harry has no abilities in that vein, the author doesn't give it much screen time. After several of the more dramatic magical battles, wizards are whisked off to Edinbourough by the likes of Listens to Wind for more specialized healing. It also worth noting that there has to be some significant healing via magic because during the war with the red court vampires and the drubbing the wizards were taking at their hands, regular mortal hospitals weren't available to the wizards as their powers would put all of the high tech equipment on the fritz, putting their own health and anyone in the hospital depending on that machinery at serious risk.

There are healers among wizards. There is no question about as it's mentioned multiple times. And they are usually eldest and most experienced wizards - which is not surprising as healing magic is very delicate and precise - and it is, as mentioned in other answers, fine line between white and black magic and thus dangerously close to breaking any of The Seven Laws. To be delicate and precise requires time, and it's also stated that not many wizards live to be old and experienced. Add to this the fact that being a wizard is also extremely rare - the whole white council probably doesn't exceed ten thousand souls worldwide (but that's just my speculation based on nothing more than hints) and you get the picture. And this also means the part of the answer Radhill gave is dead wrong - getting multiple PhDs for a being that can live for several centuries is easy. In fact, there is a description in one of the books of old wizards with dozens badges on their stoles signifying, among other things, multiple degrees - which Dresden sees during one of the White Council round-ups.

Wizards are like ordinary humans in terms of physiology - they can be wounded, maimed or killed. But, we also know that if they can get immediate or quick medical attention, they can heal from nearly everything with enough time. Thus the need for hospitals, but not exactly pressing need for medical magic. Even in classic medicine every doctor will stress the need for time to heal with proper treatment. Well, with wizards there is less treatment but much more battlefield medicine.

The same and the reverse applies to vanilla humans too - damage done to them by magic is usually irreversible and even one inflicted by first-time-magic-slinger can be, and quite often is - deadly. Thus again no real need for therapeutical magic.

So conclusion would be that with so many pressures on White Council from all directions at once (even in peacetime), there is little time and resources for things that are not even close to survival requirement and with so little utility and benefit. But as mentioned in one of the answers lot of it happens "off screen", too, so we are left to a lot of speculation.

When Leanansidhe manages to heal Harryit is used as an establishing moment to show just how much she outclasses Dresden, and at the same time explains why using magic for healing is so difficult.

Here is the relevant passage from Grave Peril (Emphasis mine):

I felt my mouth drop open. "No cut?" But there had been a nice, flowing gash in my head at some point, pouring blood into my eyes and mouth. [...] How could it have vanished?
I thought of an answer and shivered. Godmother.
"No cut", she [the doctor] said. "Something that might have been cut a few months ago/"
"That's impossible," I said, more to myself than to her. "That just can't be."
[...]
If you think I should have been happy about getting a nasty cut closed up, then you probably don't realize the implications. Working magic directly on a human body is difficult. It's very difficult. Conjuring up forces, like my shield, or elemental manifestations like the fire or wind is a snap compared to the complexity and power required to change someone's hair a different color - or cause the cells on either side of an injury to fuse back together, closing it.

In the same chapter, it is said explicitly that usually, Lea has only restricted powers outside of the Nevernever, but at the moment where she performed the magic, she was more powerful than before and later in the same book - which explains her difficulty treating him later on.