Fox has a tiny viewship compared to, say, NBC in the 1980s. Murdoch may be trying to advance his world view, but he's doing it largely to a fractured (and willing, for that matter) audience.
In the "old days", there simply weren't many choices on TV, and it was easier to control a hegemonic message, Take the Cosby Show. For much of America, that was the only black family that they, 'knew', and thusly, that is how black people should act (according to the white culture). Today, a person can watch BET, or CW, or many other channels to get different takes on "black" culture. Relating this to your post, Fox exists for people on the right to get what they perceive to be reality, and MSNBC exists for people on the left to get what they perceive to be reality. Rather, it is feeding the beast.
Rather than spreading hegemony, I'd argue that MSNBC and Fox are really just catering to a specific audience that wants their world view to be validated. The hegemonic relationship is inverse, in those instances. Giving people what they want (or think they want) isn't controlling a message. Rather, it seems to be feeding the beast.