For all his power, wizard Harry Dresden does not seem to have any spells that will heal hurts. Whenever he gets knocked about in his adventures, and he usually does, he always has to wait for his wounds to heal naturally, if they ever do. He still has to wear a glove on his disfigured left hand. Yet some wizards in his world, such as Nuccio and Ebenezar McCoy, are centuries old! How can they manage such extraordinary longevity if ordinary medicine is beyond them?
Are you talking about the books or is the show still on the air? It dissapeared from my Tivo awhile back, so I assumed it got cancelled, but the thing is whacky and sometimes just decides that it no longer feels like recording a certain show.
Traditionally, in fantasy, healers are a breed unto themselves, separate from other spellcasters. Maybe Nuccio and McCoy have someone else healing them, or maybe they don’t go around getting hurt all the time.
Alternate answer: a wizard did it.
Wizards don’t get cure spells. That’s reserved for Clerics and Bards. 
With the obligatory D&D joke out of the way, my WAG is that magic in Harry’s world is primarily about using your own energy to manipulate things. I’m only midway through the third book in the series, but he frequently uses internal energy to power his spells. Presuming the magic follows some pseudo-physical laws, it would make sense that you can’t receive more than what you give. Healing himself would probably expend far more energy than he’d get back, and would otherwise be a waste of time.
ETA: It’s also possible that he simply can’t use any magic to affect himself. I’ve yet to see him use a fly spell or anything similar. When he does need to use magic to affect himself, he brews up potions to be drunk at a later time. Who knows, maybe he does have a recipe for Cure Light Wounds, it’s just never mentioned in the books or show.
I recently read White Night and I seem to remember it being mentioned by Harry that wizards are naturally long-lived. Part of this could include better recuperative powers than “normal” people. The damage to his left hand seems to be slowly healing itself; he was told that he would never have the use of it again, and it was recommended that he just have it amputated.
Yep, wizards are different from normal humans. In Dead Beat,
his pal Butters confirms that his hand is healing, along with other old injuries. This is probably the cause of wizardly longevity.
Hey, this might prove interesting. It doesn’t answer the question you asked, BG, but it’s an explanation for why Harry doesn’t have healing spells. Book 3, Grave Peril, chapter 22 (sorry, I can’t give an exact page number):
It’s a model of magic that involves directly manipulating nature, it seems, rather than simply saying “Let it be so!”
Thanks for the OP. Cool beans, book 10, Small Favor, is out.
Off to interlibrary loan…
Aaaarrrggghhhh! 36th in the queue! 
It’s a couple of things. First of all (as others have said), it’s been established in the books that wizards are naturally long-lived, and that they are also naturally quick-healing. Harry’s hand is healing, just slowly. For a regular person it wouldn’t ever have healed at all.
Also, it’s been established that different wizards have different strengths. Harry is good, as he puts it, at “Finding things. Following things. Blow[ing] things up.” His apprentice, Molly, is good with emotional and mental magic. It could be that there are wizards whose particular strength is healing. But we haven’t met any of those yet.