The pair began sparring after Acosta asked Mr Trump about the caravan of migrants heading from Latin America to the southern USA border.
"CNN should be ashamed of itself, having you working for them", Trump told Acosta during the press conference, as the reporter battled to keep a microphone amid attempts by a female intern to retrieve it.
Acosta called the accusation that he put his hands on the intern a "lie".
Acosta, in a statement, denied Sanders' statement and said he and CNN "won't back down" after the ban. The network also said that "Press Secretary Sarah Sanders lied" about Acosta's interaction with the intern.
"The most unsafe type of fake news and reporting and evidence is when you get into the fine details, the nuanced things that are shaped to present a certain viewpoint or decision or news a certain way", he said.
The "actual fake news" hysteria started after Sanders announced that Acosta's hard pass had been revoked "until further notice". "While President Trump has made it clear he does not respect a free press, he has a sworn obligation to protect it".
The president and the reporter sparred at a press briefing Wednesday, culminating with a White House intern trying to pull the microphone away from Mr Acosta.
Acosta responded nearly immediately to Sanders' statement on Twitter, calling it "a lie".
The president then turned to NBC News' Peter Alexander, who tried to defend Acosta.
President Trump dismissed his characterization of the caravan as a "difference of opinion", adding, "I want them to come into the country, but they have to come in legally". "The video is clear, he did", she said in a statement on November 8. "The point of yesterday's session was merely to antagonize Trump's and his supporters' favorite bogeyman, the free press, in order to keep them galvanized and everyone else distracted in the wake of a stinging political defeat".
"Unfortunately this criminal will have to be dragged lying and obstructing his way out of the White House".
Reporters and historians said it was hard to recall a similar episode.
Journalists assigned to cover the White House apply for passes that allow them daily access to press areas in the West Wing. "Sped up to make Jim Acosta's motion look like a chop", Shimunov wrote.
Conservatives cheered the administration's decision to bounce Acosta, who never misses an opportunity to attack the president.
The White House Correspondents Association also condemned the action to revoke Acosta's credentials.
An activist interrupted President Obama at a White House event celebrating L.G.B.T. Pride Month to demand an end to the deportation of L.G.B.T. immigrants. "We stand by our statement", Sanders said in a statement.
The intern attempted to grab the microphone from Acosta, who nevertheless tried to shout out one last question.