Business Insider has a comparison of Fox News ratings since the White House launched its campaign against the network, and it finds Fox's audience share among cable networks has grown nine percent in the past month. CNN, meanwhile, is now last among the cable news networks in prime time viewership.
The same ratings-bump phenomenon happened for Rush Limbaugh, whose ratings have hit record highs in some markets since the White House made him a target.
Rush's bump actually started before Democrats launched their organized campaign against him, which began during the first couple days of March with a coordinated effort to refer to him as the leader of the Republican Party. Improved ratings were reported around that time--including a 45 percent gain in New York and a 30 percent gain in LA--and were posed as evidence that the White House's campaign had backfired. But the numbers were actually from February, when tensions were mounting but the White House/Democratic messaging effort hadn't yet begun in full.
Limbaugh's ratings did continue to climb after the Democratic anti-Rush campaign went into full swing, so that's not to say the White House's messaging attack didn't help him.
But Fox, similarly, has been trending upward since before the White House launched its communications war: according to TV-ratings blog TV by the Numbers, Fox's primetime ratings climbed steadily over the summer to approach levels seen during the heat of the 2008 election--when viewership and readership peaked for all political media outlets--whereas CNN and MSNBC's ratings had fallen off since November.
See TV by the Numbers' graph below:
Perhaps it's just a facet of the post-election Obama era that a larger market share goes to conservative punditry, which Fox features in its primetime lineup, as Obama's supporters breathed a satisfied sigh of relief and trusted the new president and his agenda, while Obama's critics have felt more of a need to stay on their toes.