Specifically, women being more likely to initiate divorce has nothing whatsoever to do with what the parent is saying. The reason behind that statistic is that women are no longer as dependent on men and therefore have more freedom to end the marriage if they aren't happy with it.

Women have a lot more invested in sex and children than men do. And therefore, they are looking to have a stable situation to have a family more than men are. That situation was typically marriage, but now with the state being able to take care of the children so much of the time (public school, various welfare programs etc.) there are lot more single mothers. I think single motherhood is highly correlated to the amount of governmental assistance in a neighborhood. To the extent that there is single motherhood in an area, you may be right -- women may be much less willing to make a relationship work.

As an aside, I am not sure we think that's healthy or desirable. Children born into two-parent households have been shown to do far better on average than children born into single parent households.