Obamacare started its life out as a pejorative term used by those critical of the new Health Care Law. But, as these things go, it's not so bad--nothing like as bad as "death panels," for example.
So the administration isn't fighting or ignoring the use of this term. It's embracing it and recognizing it as an opportunity to promote its point of view. A Google search for Obamacare now produces a sponsored search result connecting to healthcare.gov which lays out the case for the new legislation. Opponents are up in arms complaining about the use of taxpayer money to buy these ads but what they're probably most upset about is the fact that their name has been hijacked.
Meanwhile, over at Yahoo!, a sponsored results battle is on. As well as the Healthcare.gov ad, there's also a link to Demand Obamacare Repeal. A new front has opened in the never-ending political battle.