The rape claim stems from an accusation Trump's then-wife Ivana Trump leveled at her husband during divorce proceedings in the early 1990s, an allegation she walked back Tuesday.
"He's speaking for himself. He's not speaking for me, obviously," Trump said to CNN's Don Lemon Tuesday, which aired on "The Situation Room."
Trump called The Daily Beast a "joke" desperate to remain relevant.
"Michael was extremely angry because he knew (the alleged incident) never took place," Trump told Lemon.
Marital rape has been illegal in all 50 states since 1993 and non-consensual sex between spouses does in fact constitute rape, said Scott Berkowitz, the president and founder of the Rape, Abuse & Incest National Network.
Cohen apologized for his comments Tuesday, calling them "inarticulate," and said the reporter's question sent him into a tailspin.
"As an attorney, husband and father there are many injustices that offend me but nothing more than charges of rape or racism. They hit me at my core. Rarely am I surprised by the press, but the gall of this particular reporter to make such a reprehensible and false allegation against Mr. Trump truly stunned me. In my moment of shock and anger, I made an inarticulate comment - which I do not believe -- and which I apologize for entirely," Cohen said in a statement to CNN.
Cohen, who is one of Trump's top lawyers, threatened to sue the Daily Beast reporter and ruin the reporter's life.
"I will make sure that you and I meet one day while we're in the courthouse. And I will take you for every penny you still don't have. And I will come after your Daily Beast and everybody else that you possibly know," Cohen said, according to the Daily Beast. "So I'm warning you, tread very f---ing lightly, because what I'm going to do to you is going to be f---ing disgusting. You understand me?"
"Mr. Trump didn't know of his comments but disagrees with them," Trump campaign manager Corey Lewandowski told CNN, referring to Cohen's remarks to the Daily Beast.
In a statement obtained by CNN, Ivana Trump said the Daily Beast story "is totally without merit."
"I have recently read some comments attributed to me from nearly 30 years ago at a time of very high tension during my divorce from Donald. The story is totally without merit. Donald and I are the best of friends and together have raised three children that we love and are very proud of. I have nothing but fondness for Donald and wish him the best of luck on his campaign. Incidentally, I think he would make an incredible president," Ivana Trump said in the statement, which was verified by the Trump campaign.
The campaign also distanced itself from Cohen, who has spent weeks appearing on multiple television news shows, including CNN's "New Day," to play up Trump's candidacy for president and defend the campaign from attacks from Trump's primary opponents.
"Michael Cohen is a corporate employee and is not affiliated with the campaign in any way.=," Lewandowski told CNN.
When asked whether Cohen would continue to make television appearances and behave as a surrogate for Trump, he dismissed the question, saying the campaign and Trump's company are separate enterprises.
"This would be like asking Nike what they think of Reebok," Lewandowski said.
"Mr. Trump speaks for Mr. Trump and nobody but Mr. Trump speaks for him," a campaign source told CNN on Tuesday morning.
A second campaign source toed the same line and pushed back against the notion that Cohen is a surrogate for the campaign.
"He is speaking as someone who has great insight into Mr. Trump's skills as an executive," the source said.
Both sources emphasized that Cohen is employed by the Trump Organization and not the campaign.
Cohen has not only repeatedly appeared on TV to support Trump's presidential campaign, but he has also provided statements in response to political reporters' inquiries about Trump campaign controversies. Cohen sent CNN a statement via email earlier this month when Trump's official Twitter account posted photo of men in Nazi uniforms.
In a deposition during divorce proceedings, Ivana Trump accused her husband of raping her during a 1989 incident, an accusation that was first revealed in the 1993 book by former Newsweek reporter Harry Hurt III, "Lost Tycoon: The Many Lives of Donald J. Trump."
CNN could not obtain a copy of the deposition.
As the book was about to be published, Ivana Trump wrote a statement that was printed on the first page of that book:
"I felt violated, as the love and tenderness, which he normally exhibited towards me, was absent," she said in the statement. "I referred to this as a 'rape,' but I do not want my words to be interpreted in a literal or criminal sense."
A Trump campaign spokesperson said in a statement that the rape accusation "is old news and it never happened."
"It is a standard lawyer technique, which was used to exploit more money from Mr. Trump especially since he had an ironclad prenuptial agreement," the spokesperson said.
Democrats quickly pounced on Cohen's remarks, with Democratic National Committee Chairwoman Debbie Wasserman-Schultz calling Cohen's comments "a new low" in a statement Tuesday morning and telling Republicans to take a stand.
"Rape is rape. Full stop. End of story. There is no difference or division between 'forcible', 'legitimate', 'marital' or any other label Republicans slap on before the word 'rape'. All rape is a disgusting violation, and Americans have fought too long and hard for that to be acknowledged to still have it questioned in 2015," Wasserman-Schultz said in the statement. "It's a pattern of outrageous comments that must stop, and Republicans should call it what it is - despicable."
Ivana Trump is Donald Trump's first wife and mother to his three oldest children: Ivanka, Eric and Donald, Jr. who are all top officials at The Trump Organization. Trump is now married to his third wife, Melania Trump.