President Trump recently printed this:
The FAKE NEWS media (failing @nytimes, @NBCNews, @ABC, @CBS, @CNN) is not my enemy, it is the enemy of the American People!
and there are a large number of similar statement he's made on the subject.
I'm just wondering if the media outlets can prove a loss in correlation with his statements, and he can't back them up, if big media companies can sue Trump or even the US government (not sure how affiliation works there).
We may soon have a more definitive answer. A Grand Junction, Colorado newspaper is suing a politician for calling it "fake news", and the resolution of that case and the hypothetical that you propose would turn on the same legal principle.
It is highly unlikely that such a lawsuit would prevail, because "fake news" probably doesn't constitute libel per se, because the comment could be construed as hyperbole or as a statement of opinion (neither of which are actionable), and even potentially, because a "speech and debate clause" defense under the state constitution might apply (depending upon the context in which the statement was made by the politician). The context of the particular tweet cited generally defines specified organizations as "the FAKE NEWS media" rather than accusing them of any particular instance of making a false statement, so it is probably an opinion or hyperbole.
But, if the statement were made knowing it was false or with reckless disregard as to truth or falsity, and if the term "fake news" in the context in which it was used could be legitimately construed the imply a statement of fact which is not true, it wouldn't be impossible for the lawsuit to succeed, and depending upon the context of the statement, it could have such an implication.
A suit against Trump could also implicate Presidential immunity doctrines which are more robust than immunity doctrines for other public officials, particularly if the "fake news" comment could be construed as part of the official duties of the President (for which there is absolute immunity) as opposed to his unofficial duties. The immunity question is a closer one than the question on the merits of defamation law about which there is much more case law to flesh out what is and isn't covered.
"actual malice":"knowledge that the information was false" or that it was published "with reckless disregard of whether it was false or not."seems like someone could make a case regarding reckless disregard