the question is whether the behaviour differences in men and women are natural or cultural

Obviously both, but not in a way it is often shown.

We have different bodies and capabilities, different needs. So our experience of same events would be different, and from different experience different behaviour and character are shaped in the end.

Some role are playing natural instincts (protecting kid by women) and hormones (progesterone is sedative and light painkiller, testosterone is increasing sex drive and agression). That is one of reasons why men are so much more hysterical in general (thought they are showing this in violence or "fighting back" what makes them uncomfortable, reacting to much smaller problems as they are huge ones) and less tolerant to physical pain, thought male body is much faster regenerating injuries because of faster metabolism, more oxygen in blood, faster blood flow, etc. Sorry for using word "hysterical", I just don't know another word in English to describe the situation. In English this word is pretty bad one, as it was mostly used to misdiagnose women in the past by saying "your health issues are just tricks of your mind and hysteria".

However, in social cultures, all those differences can be ignored, as socialization can completely change them all. Social mammals have great capacity of "going against nature", we can see that in dolphins and monkeys. And even among smart birds like crows it happening, when they are going against their instincts if it is what is the norm in their local group.