Now, what Quayle said was:
“Bearing babies irresponsibly is simply wrong... Failing to support children one has fathered is wrong. We must be unequivocal about this. It doesn’t help matters when prime-time TV has Murphy Brown, a character who supposedly epitomizes today’s intelligent, highly paid professional woman, mocking the importance of fathers by bearing a child alone and calling it just another lifestyle choice.”Quayle took some gratuitous shots at women in his notorious remark, and he could just as easily have put his main point in a way that nearly everyone would agree with. So, I dislike this simple declaration that he was right (which I've seen many times). He phrased it in a way that's either deliberately or unwittingly provocative.
Some single-parent families are better than others and some mother-and-father families are worse than others. I question whether it's the single parent per se that's the problem — where it can be shown that it's worse — or whether more women who get into the single-parent position are the irresponsible/disagreeable/poorly skilled kind of person.
Another problem with what Quayle said is that he characterized the kind of woman that we'd presume to be responsible — the "intelligent, highly paid professional woman" — as doing 2 things she might not have been doing: 1. "mocking the importance of fathers," and 2. treating child-bearing as "just another lifestyle choice." This "Murphy Brown" woman might very well prefer to have a solid marriage with a father in the house but not be able to make that happen. And she might dearly and virtuously want to mother a child.
I know Quayle was saying that the TV character provided women with a role model and might have fooled us into thinking raising a child alone would be easier than it is, but he was focused on getting out a rival moral message. Murphy Brown is telling you it's fine, but I need you to know it's wrong. There's a difference between saying it will be hard and it's simply wrong.