I think... women need men about as much as men need women. No more, and no less.
That is, most people are happier if they're in a relationship (or, at least, getting sexed up on the regular), and most people are at least mostly heterosexual, so most women are better off with a man in their lives--but most men are also better off with a woman in their lives.
And, since men and women have (on average) different strengths and abilities, a household with both a man and a woman will be more likely to have all necessary skill sets and abilities covered than one with only one person, or with two men or two women. It's not guaranteed, either way, but... it tends to help.
And, of course, if there are children involved, two parents are generally better than one, and (because of the different abilities thing, as well as having both male and female role models), 2 opposite-sex parents are a bit better than 2 same-sex parents, all else being equal.
But, well, women have no greater inherent need for men than men have for women. Women don't "need a man around the house" to take care of them, or to tell them what to do, or whatever. An unmarried woman is no more likely to be unhappy, unhealthy, or otherwise in bad shape than an unmarried man, and possibly less so (statistically, married men live longer than single ones, but single women live longer than married ones). And so on. The saying "A woman needs a man like a fish needs a bicycle" came from, well, societal attitudes that suggested that women who didn't have, and weren't in the process of obtaining, A Man, were somehow defective or broken or whatever, while being a "confirmed bachelor" was seen as a perfectly valid and acceptable choice.