In a series of tweets before dawn Monday, he called Streep, who has more Academy Award nominations than any other actor in history, "one of the most overrated actresses in Hollywood."
"She is a Hillary flunky who lost big," Trump said.
At the Golden Globes on Sunday night, Streep used an acceptance speech for a lifetime achievement award to take Trump to task. Without naming him, she spoke out against disrespect and violence, and called for the protection of journalists.
She specifically called out Trump's 2015 mocking of a reporter with a disability.
"It was that moment when the person asking to sit in the most respected seat in our country imitated a disabled reporter," she said. "Someone he outranked in privilege, power and the capacity to fight back."
Streep's speech wasn't unusual for an awards show -- many actors who've won an award have used the forum as an opportunity to air their thoughts on political matters. There's a long history of that, from 1973, when Marlon Brando declined the Oscar for best actor, which he won for his role in "The Godfather," and sent activist Sacheen Littlefeather in his place to protest the treatment of Native Americans, to 2015, when Patricia Arquette used her best supporting actress Oscar win for "Boyhood" to draw attention to gendered wage inequality.
But Sunday night's speech was amplified by how big a star Streep is -- big enough that she's now famous even for how good her acceptance speeches are -- the way she delivered it without even mentioning the name of her target, and the general mood of the country.
Trump claimed on Monday, as he has in the past, that he never mocked the reporter, Serge Kovaleski of The New York Times. Video shows he did,
flailing his arms in imitation: "You gotta see this guy."
"For the 100th time, I never 'mocked' a disabled reporter (would never do that) but simply showed him "groveling" when he totally changed a 16 year old story that he had written in order to make me look bad.," Trump tweeted. "Just more very dishonest media!"
Trump's skirmish with Streep comes at the start of an important week in Trump's transition into the highest office in the land, with hearings for multiple top nominees and an eagerly-awaited press conference on the schedule.
It is yet another example of Trump engaging in a culture war against the industry that helped to make him famous. These examples often come when Trump is criticized or shunned by the celebrities he tried for decades to embrace.
Last month, after reports that his camp was having trouble attracting celebrities to his inauguration, Trump similarly lashed out on Twitter.
"The so-called "A" list celebrities are all wanting tixs to the inauguration, but look what they did for Hillary, NOTHING," he tweeted. "I want the PEOPLE!"