I think a reason for the phenomenon you described is that men in many cultures are encouraged to be "tough" and to act as a strict alpha figure in any social group or situation more so than a woman. He is also expected, according to Western and Eastern cultural fiats, to come off as a "protector" of the family, and to fend off rival males to the various aspects of his social status. After so many generations of this type of social indoctrination, it explains why men are not only quicker than women in general to resort to knee-jerk violent solutions to any problems or challenge, but because they feel their social reputation demands that they exhibit physical "toughness" at every opportunity. This also explains why sports with a visceral physicality are more popular among boys/men than girls/women, and manifests today with the popularity of violence-laden video games with males to a greater extent than females.

Women, in contrast, have long been conditioned to come off as "delicate" and nurturing, and to consider it "unladylike" to resort to violence, depending on men to protect them from danger instead. This had (and still has) obvious advantages to the patriarchal hierarchies of the world, as women who were less overtly "tough" than the men, as well as dependent upon them for protection and support, were far less likely to challenge the gender-based hierarchy. This resulted in a double-edged sword of women often being greatly coddled and believed to be inherently morally superior than men, while simultaneously awarding the bulk of social, political, and economic power and decision-making to men.

Of course, once women enter the same realms of power that men do, including politics, they became as quick to authorize violence or a "show of force" as any man does. This implies that a large component of the proclivity towards violence may have more to do with perceived status than actual physical gender. Note not only my example with politics, but also girls who are part of street gangs, either all-female or mixed gender.

There may be some genetic components to it, but I think its extreme manifestations in Western and Eastern cultures are largely learned, and harken back to the days when men were required to perform most physical and risky responsibilities, including warfare and exploration. This was because life was more precarious then; and since women were seen as inferior in a social context due to the men being the "protectors" while simultaneously more valuable than men for the preservation of the tribe and even entire human species since they were the physical child-bearers.