I wrote up the following response and then realized that I was answering a different question than the one you asked. The following is an investigation of whether the tendency for child-raising to be "womens' work" is caused by genetic or cultural factors. This isn't directly related to the question of whether the personality traits are caused by genetic or cultural factors. I'm posting it anyway because I spent an hour writing it up and don't want to throw it away.
I did some quick research and found that even in "matriarchical" societies, women are usually more involved in infant care than men. I started with this article: <http://metro.co.uk/2013/03/05/where-women-rule-the-world-mat... and went down the list:
* Regarding the Mosuo, this Wikipedia article: <http://en.wikipedia.org/wiki/Mosuo> writes that "Children of such relationships are raised by their mothers and the mothers' families." However, men are involved in raising their nieces and nephews.
* While men are heavily involved in child-raising in Aka societies, this source: <http://books.google.com/books?id=21z2hbOuTcEC> writes that "The Aka are exceptional in comparison to other societies in that fathers are actively involved with infants and are second only to mothers in the amount of direct care to their infants." (pg. 5)
* Despite the existence of sworn virgins, Albania is patriarchical, according to this article: <http://en.wikipedia.org/wiki/Albanian_sworn_virgins>
* I wasn't able to find sources regarding childcare in Iceland or Meghalaya.
In addition, the fact that there are so few matriarchical societies in the world is notable in and of itself.
My hunch is that the reason why women are more involved in childcare than men is due to cultural factors which are caused by genetic factors. One could certainly imagine a culture in which as soon as a woman gives birth, she hands the infant off to the father, and then has no further involvement in its care. So there is no immediate reason why women must be the caregivers. But that hypothetical society would require that babies be bottle-fed, which is a recent invention; historically, children had to be breast-fed, and genetic factors are certainly responsible for the fact that women are much better than men at producing milk. So the women would have had to be involved in caring for their children at least at the beginning. This could lead to a cultural norm for women to remain involved in childcare even after it is no longer necessary to breast-feed.
Of course, that's not to say that childcare should be "womens' work". Even if women do turn out to have a genetic propensity towards childcare, that doesn't mean that we as a society have to let that dictate our gender roles.