More women than men care about height preferences/requirements, with a little over half of men caring, but there being at least 94% of women who outright refuse to date a man shorter than themselves.
This is due to sexual selection; women want to pass on the 'fittest' genes to their offspring. Females are the selectors (outside of rape), by carefully selecting which men they mate with, they guard which genes get passed on to the next generation, and which don't.
For every inch that a man becomes shorter, his dating pool becomes exponentially smaller. An extremely short man has an extremely small dating pool. An extremely short woman has more options, because short girls are considered feminine and cute, and men are generally much less selective in their choice of sexual partners.