One of the favorite techniques of some commentators on the right is to say really outrageous, hyperbolic things, things they often don't really even mean, in order to draw criticism and outrage from the media or politicians. Then they point to these things and say 'See how they hate us?' Ann Coulter and Bill O'Reilly are masters of this technique, and Glenn Beck is getting pretty good at it.
I don't believe Obama has ever criticized Fox, just his people--spokesmen and press secretaries. What they said was that Fox was an arm of the GOP, and I think that's pretty much obvious to everyone on both sides. It would be more specific, and more true, to say that Fox was nothing more than a propaganda organ for the Republican Party. They don't even TRY to be 'fair and balanced'. Their motto should not be 'We Report, You Decide', it should be something more like 'We Give You Our Side of the Story, and You Swallow it Whole'. 8^)
Republicans have been feuding with the mainstream media for decades now, starting with President Nixon. Nixon, Reagan, and both Bushes -routinely- excluded newsmedia they didn't think gave them a fair shake, or who reported news they found embarrassing. But this is just another item of that long list of things that are wrong or criminal only when Democrats do them.
And you're right, nobody wanted to deny Fox their 1st Amendment rights. It's not a war on Fox, it's a war on lies.