Source: swiped verbatim from Time.com's weekly web feature, Verbatim
While it's true that none of us know the context of this quote, given that past GOP majority leaders (::coughTrentLottcough::) haven't exactly been known for their inclusive mindsets, I think its a fair assumption that Delay just held up a big "I'm a sexist" sign.
Note the wording of the quote. A woman "can take care of the family", but structure and stability are the provisions of "a man". These notions about gender roles which are more than a little antiquated. Now, I'm very much a proponent of a man sticking around and being a father. In the same way I'm very much a proponent of a woman sticking around and being a mother. I do believe that a lot of the problems we see with kids, today, stem from the absence of one, the other, or both parents. But to proport that a man/woman is incapable of both taking care of and providing a structured, stable home for his/her family is sexist. Furthermore, it implies that there are certain roles into which Delay would see men and women pigeonholed. The woman is the caretaker and the husband the provider. This is the traditional conservative image of the American household. More and more families are differing in their makeup. In some, for example, the father is the nurturer and the mother the provider.
So, while I agree with Delay that the presence of both parents aids in the maintenance of a happy, healthy home, I disagree with him regarding the roles which a woman plays and which a man plays. There should be no pre-determined role for either the mother or the father. Furthermore, I disagree with him that, in the case where one or the other parent is absent, that the present one will not be able to act as both caretaker and provider, be that parent the mother or the father.