After months on the defensive over his health law, a more combative President Barack Obama has emerged to fight about gender politics, leading to an election-year competition with Republicans for support from women. No single group will be more important to Democrats’ fortunes, say White House advisers, than unmarried women, who are likely to go Democratic—if they vote, and that’s far from certain when trust in Washington is low. The president is trying to convince women that Democrats are more concerned about improving their financial standing in difficult economic times, and he charges Republicans with standing in the way. “Republicans...