Correct. We don't choose our chromosomes. They determine whether we'll be a woman or a man. This happens at the moment of conception.
Woman is the word for a female human.
Man is the word for a male human.
In other species, for instance:
Sheep - Ewe, Ram
Cattle - Cow, Bull
Rabbit - Doe, Buck
Chicken - Hen, Cock
Fox - Vixen, Dog
None of those words refer to psychological characteristics, social behaviours or personalities. They're just terms to distinguish the two reproductive categories within a species. The females of some species are extremely aggressive and violent; it doesn't mean they are male. Their reproductive category is female: the egg-producing sex. Some males are docile. As they're of the sperm-producing category, we don't change their species-sex designator to 'doe' or 'hen'.
In exactly the same way, a sweet-natured male human is a man. He was destined to become a sperm-producer when the egg absorbed a sperm with a Y chromosome. All this means is that he'll probably produce sperm as an adult, and will definitely not produce eggs. He will have the physical configuration of a human male, the strengths & vulnerabilities of human males, within the range associated with his sex. He's a sweet-natured man. That's it.
To believe a gentle, passive, caring nature defines a woman is not only grossly reductive and sexist; it's also plainly false. It would mean his personality altered his sex from male to female. His body can't produce eggs. His Y chromosome destined him to produce sperm - whether he can or not, he's of the male sex. And the word for human males is 'men'.
Personally, I'm really glad that docile men and aggressive women exist. Part of the glorious variety of human life is that we all have the full range of psychological potentials, each of us with an individual mixture. I think that's worth celebrating.