In a nutshell, he uses Scandinavia (and more in particular Norway) as an example of a social experiment where the high level of equality between men and women has seen a stronger distinction between the gender presentations instead of the reverse. Women appear very feminine (almost Barbie doll like) and men appear quite masculine.
He argues that if gender identity were socially based then you should be seeing more of a blurring between the sexes instead of a sharper distinction and argues that Norway, as one of the most gender egalitarian countries on the planet, provides proof of how much core gender identity is primarily biologically predetermined.
Inherent in your sense of gender is also the aspect of being sexually desirable which is why many gender conflicted people (myself included) experience what Ray Blanchard termed autogynephilia.
I include Jack’s main conclusion:
“Norway can be understood as a social experiment aimed at determining whether there is a biological core to your sex identity.
If "gender identity" was socially constructed only, you would expect to see a convergence of male and female gender expression in Norway. What you see is the opposite.
For me this is a clear indication -- although not proof -- of a deep felt need to be read as a man or a woman. I suspect that this need is related to sexuality in the broader sense: to sexual desire and the longing to be desired. This need is not limited to heterosexual persons.
If there is such a basic sex identity, it is clear that it is not defined by temperament, abilities or other personality traits. If there are any differences at all between men and women in this respect, they are marginal and only valid on an aggregated statistical level.
Nor is this sex identity defined by specific symbols or ways of behavior. Men and women make use of the symbols they have at hand. There are many cultures where long loose hanging hair and a narrow waist would be considered a sign of masculinity.
Moreover, this instinct varies in strength. There is always a minority of people who feel unease when asked to choose between being male or female.
This leaves us with an instinct that entices most boys and girls, men and women, to express their sex identity with whatever symbols the surrounding culture may provide. Such an instinct could be compared to the hunger instinct. The feeling of hunger encourages you to eat, but the preference you may have for particular types of food is cultural.
This sex identity instinct would explain why kids at a certain age reduce their playtime with kids of the opposite sex. They make boy-clans and girl-clans in order to be able to test out their roles as boys and girls. The Barbie look of the modern Norwegian woman could be understood as an extension of these games.
This would also explain the need many transgender feel to make use of the symbols of their target sex. If this is the case, crossdressers crossdress in order to express an inner sex identity. In this context a male to female crossdreamer's urge to dress up as Barbie makes perfect sense. The crossdresser is doing exactly the same as Norwegian XX women do.
In such a model it becomes impossible to take the sex out of sex, in the sense of distilling a kind of "womanhood" that is totally devoid of sexual desire. The drive towards recognition as a woman definitely has a strong sexual component. But that does not mean that you can reduce femininity (transgender or non-transgender) to sexual desire -- no more than French cuisine can be reduced to a count of calories..."