I don't watch Fox News, as, when I watch cable news, I do so early in the morning, and Fox at that time shows only the annoying "Fox and Friends." I don't watch opinion shows of any kind, on Fox or other networks. What little I've seen of them disappoints, regardless of the political point of view pontificated. Olbermann seems patronizing; Beck is simply creepy. CNN's Headline News is tolerable, as commentary is largely absent. I'm indifferent to pundits generally.
I think, however, that it is no longer reasonable to classify any media as objective, if ever it was. Technology now allows and even encourages the proliferation of opinions, regardless of merit. Media seems compelled to produce supposed experts to comment and express opinions regarding virtually anything, on any occasion, and at all times.
If, therefore, there is anything special about Fox News, it's likely that it is less subtle in promoting its point of view than other networks, or that it does so more often than others.
That the President's minions seek to detract from the status and popularity of Fox News is not surprising. The Administration has its own views, and wants those views to be accepted. People are, I think, free to boycott it or any other network. Expression of displeasure, or even contempt, doesn't amount to actionable limitation on the "freedom of the press" which some, like Gore Vidal, would claim has not been a factor in our republic for quite some time in any case.
I think the administration can only take this so far, however. I think there are some in the media whose reaction to the President is extreme, and peculiar. This should be noted. But it ultimately will do him no good to engage with the mudslingers on their level.