Chiroptera writes:
Notice the phrasing: "knowingly publish false information". And this is a pretty general statement. I'm guessing that any specific proposal to enact such a thing would end up with little support.
It should be unnecessary for journalism as a whole. Legitimate media outlets will fire or suspend journalists for purposefully pushing false stories. Journalists can even be fired for mistakenly reporting false news (e.g. Dan Rather). Again, this is at legitimate media outlets. The best way to determine if a media outlet is legit is to see how they handle stories that turn out to be false.
I don't think the danger here is that US citizens will support restrictions on the press. The danger is that a sudden and temporary panic may cause US citizens to allow dangerous legislation to be enacted without proper thought and debate.
"Hoisted by one's own petard" comes to mind. What happens when people's favorite commentators over at Fox News start getting sued and jailed for reporting nonsense?