I would re-title this to: "Republicans Winning the War on Repubilcans." Because that's what "The War on Women" really is.  Here's the money graph:

More voters think Mitt Romney and the Republican Party respect women who work outside the home than think President Obama and the Democrats respect women who stay at home, according to the latest The Hill Poll.

Forty-nine percent of likely voters said the presumptive GOP presidential nominee respects women who have independent careers, while 27 percent said he doesn’t and 24 percent weren’t sure.

When asked if President Obama respects women who stay at home rather than pursue a career, 37 percent of likely voters said he doesn’t and 35 percent said he does. Twenty-nine percent were unsure.

And as this piece points out, President Obama benefits more from a "marriage gap" then a gender gap. He's popular with single women. Married women is a different matter. I'm guessing Team Obama's polling already told them this, which is why Hilary Rosen had a democrat brick house fall on her.