The Republican said Christine Blasey Ford should be heard but that her testimony would not likely change his mind about Brett Kavanaugh.
"The nomination must be withdrawn," said Avenatti, amid evidence he claimed to have of Supreme Court nominee Brett Kavanaugh having plied women with drugs and alcohol while in high school.
CNN's Stelter lashed out at right-wing pundits.
The Truth and Courage, or TNC, society has since disbanded.
The Senate Judiciary Committee released the letter, which originally had been sent to Senator Dianne Feinstein in July.
The Republican questioned how such an investigation would be carried out.
Attorney Michael Avenatti suggested Sunday that Dr. Christine Blasey Ford may not be "alone" in her accusations against Supreme Court nominee Brett Kavanaugh.
The panel of GOP women suggested they would back Kavanaugh regardless of whether the claims were true.
Garrett Ventry, a communications aide to the Senate Judiciary Committee, denied the allegations.
Without an investigation, Kavanaugh's confirmation would undermine the public's faith in the Supreme Court, according to the open letter.
"Judge Brett Kavanaugh is a fine man, with an impeccable reputation, who is under assault by radical left wing politicians," Trump tweeted of his Supreme Court nominee.
President Donald Trump said he believes "everything's going to be just fine" with Supreme Court pick Brett Kavanaugh's nomination.
Representative Ralph Norman of South Carolina mocked Ruth Bader Ginsburg's age, and sexual assault, in suggesting she was groped by Abraham Lincoln.
The journalist said Ford should testify on Monday.
Hollywood actors and women from across the country are showing support for Christine Blasey Ford.
The president claims Democrats timed the revelation of the allegation to obstruct the Supreme Court nominee's confirmation.
The former secretary of state was asked about due process over allegations.
Christine Blasey Ford, who has accused Supreme Court nominee Brett Kavanaugh of sexually assaulting her in the 1980s, has been forced out of her home after receiving death threats, her lawyers say.
The president defended the Supreme Court nominee's character against allegations of sexual assault.