In our social setting, many women see the role of healer as natural, appropriate, expected. And men are equally conditioned as wanting to fill the role of the protector as well as the leading actor.This is quite possibly true. In fact, among Asian WoW players, the gender differences are even starker, perhaps reflecting a society where the traditional gender roles are stronger.
However, I'd like to propose an alternate explanation. Note that Milady does not consider the DPS classes. But there is a gender divide in the DPS classes. Female players are more likely to play ranged dps classes such as warlocks, mages, shadow priests, etc. than melee classes like rogues and warriors.
If you add the ranged/melee split to the tank/healer split, I think there's a stronger factor present than gender roles: range. Healers and ranged dps operate at a distance from the enemy, while tanks and melee dps have to get up close.
So why choose stay at range? One possible explanation is risk-aversion. As a group, women are normally more risk-averse than men. So it makes sense that female players would gravitate to the classes that can stay away from the enemy the longest. Even healers fit into this scenario better. If you want to avoid dying, what's better than a class which can restore health?
Sadly, we don't really have access to good data that identifies the gender of the player. There are some interesting questions that we could look at. For example, among female DPS shaman, what proportion plays Enhancement (melee) as opposed to Elemental (range)? The same question could be asked for Druids: Feral or Boomkin?
What about Monk healers? Do female Mistweavers "fistweave" or do they heal in the traditional style? Are women who tank more likely to tank on their second character, once they have a better handle on the game and the risks involved?
If we just looked at the healers, is the proportion of female paladins higher or lower than the norm for the other classes? Perhaps risk-aversion thesis should predict lower paladins, because paladins are traditionally melee. Though I suppose this is countered by the whole hard-to-kill vibe with the bubble and plate armor. I'm not entirely sure if a risk-averse person would be attracted to paladins or not. Plus, we'd have to account for the general awesomeness of paladins.
In any case, either theory might be true. Women and men might be responding the social pressures corresponding to traditional gender roles. Or women may be attracted to classes which appear to be less risky and more likely to survive. Or maybe both theories are in play, with different individuals having different motivations.
However, I prefer the risk-aversion theory over traditional gender roles because it also explains the bias in DPS class choice, and not just the tank/healer split.