The shocking exchange came as Acosta was asking the president about the migrant caravan.
Sarah Silverman, host of I Love You, America, also went off.
In response, White House Press Secretary Sarah Huckabee Sanders distributed a video which she claims shows Acosta pushing a female White House intern as she attempts to take the microphone from Acosta.
"These frames do not appear in the original C-SPAN footage, and appear to exaggerate the action of Acosta", Storyful's analysis said. "I do think that they're trying to shut us down to some extent, inside the White House press corps".
The president then moved to address the next questioner, NBC News White House correspondent Peter Alexander, who defended Acosta.
Looking back at Acosta, Trump added: "The way you treat [press secretary] Sarah Huckabee is disgusting, and the way you treat other people is awful".
'The way you treat Sarah Huckabee is awful, ' he said. Acosta made a decision to ask Trump why he seemingly "demonized" the immigrants traveling to the American border at the moment.
Virat Kohli makes controversial, immature comments
It is a comfortable bubble and that is why famous people must try hard to prevent it from forming. Some might say that there is so much more at stake for them given that Kohli is India's captain.
The site, InfoWars, has been a longtime supporter of Donald Trump's, and Trump has reciprocated the admiration with an interview when he was running for president. The way you treat Sarah Sanders is disgusting. "Its deceptive, unsafe and unethical", said association president Whitney Shefte.
"As working professionals, members of the news media have an obligation to verify the accuracy of what they report, with loyalty only to their readers, listeners and viewers and not to any political party, affiliation or ideology".
"President Trump believes in a free press and expects and welcomes tough questions of him and his Administration", Sanders said in her statement.
The White House's treatment of Acosta drew wide rebuke from press freedom and journalism organizations. "You run CNN, and if you did it well, your ratings would be higher", Trump said.
"I'm not a fan of yours either". CNN stood with us [when Barack Obama declared war on Fox] ...
It's unusual for the White House to pull the so-called hard passes from journalists.
"What you said is so insulting to me", he said.
Trump also told April Ryan of the Urban National Radio Network to "sit down" and wait her turn. "Even if President Trump doesn't intend it, some see him as directly appealing to the racists", she wrote. He said she was rude for interrupting another reporter, though he did briefly answer one of Ryan's questions.