Obama's war on Fox News is just plain stupid. It is, of course, utterly without merit. Fox News offers reportage and shows featuring opinion journalism and interviews hosted by conservatives. The news hours are fairly straightforward and even the opinion shows tend to feature a diverse set of guests. In this, I fail to see any distinction between Fox and, say, MNBC where the hosts tend to be liberal. Keith Olberman and Rachel Maddow are not any more "mainstream" or "objective" or "civil" than the Fox hosts.
But put that aside. You have a reputation for being coddled by the media. You are increasingly obtaining a reputation for being prickly when challenged? Why would you underscore those impressions by picking a fight with a news network who is "too critical"?
And why would you create a martyr? People in the TV news business will tell you that the Fox audience is far more politically diverse than that of the other cable news networks. Maybe that's why the administration feels the need to marginalize it, but isn't the greater danger that you will simply seem paranoid and imperious. Maybe "Nixonian" doesn't have the freight it once did, but the character flaws that it stood for are still very unattractive.