Broadcaster CBS News also voiced its intention to file a motion supporting CNN, saying it did not think taking away the press pass was "an appropriate remedy for a disagreement the White House may have with a particular reporter".
US District Judge Timothy Kelly ordered the Trump administration to respond by 11:00 (16:00 GMT) on Wednesday and set a hearing for 15:30.
And "even if" CNN could show some First Amendment harm "in the abstract", Acosta is free to "practice his profession and report on the White House", without the access that a White House press pass provides.
Asked if the White House could pull a journalist's credentials if the President disagreed with that journalist's reporting, Burnham insisted, "as a matter of law... yes".More news: Dez Bryant lands one-year contract with surging New Orleans Saints
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Acosta pushed back with more questions and a White House intern tried to take the microphone from the CNN journalist - an incident the Trump administration characterised as misconduct against the woman. That came just hours after a live on national TV dust-up between the President and Acosta at what was supposed to be a victory lap East Room press conference by Trump and descended into a verbal brawl.
In a joint statement Wednesday, the news organizations said, "Reporters covering the White House must remain free to ask questions".
Meyers went on to point out CNN's predilection for booking people whom he called "liars" on the network to sing the president's praises - and seemed to point out two of said "liars" with an on-screen graphic in the process. He persisted in questioning President Donald Trump, who lambasted the correspondent as a "rude, awful person".
White House press secretary Sarah Huckabee Sanders called the lawsuit "grandstanding". "I think there's very little to be lost by pursuing this", said Douglas E. Mirell, a specialist in the First Amendment for Greenberg Glusker.
"The President and White House possess the same broad discretion to regulate access to the White House for journalists (and other members of the public) that they possess to select which journalists receive interviews, or which journalists they acknowledge at press conferences", lawyers say in the filing.
Washington Post chief executive Fred Ryan said: "We support CNN in its effort to restore the press credentials of its White House reporter".
CNN has some 50 other employees who still have White House passes, the president's lawyers said.