By Clare Foran, Ted Barrett and Devan Cole, CNN

(CNN) -- Top congressional Democrats are calling for Labor Secretary Alexander Acosta to resign over his handling more than a decade ago of a plea deal for multi-millionaire Jeffrey Epstein amid a debate on Capitol Hill over whether Congress should investigate the agreement.

Federal prosecutors in New York unsealed a criminal indictment on Monday charging Epstein with having allegedly operated a sex trafficking ring in which he sexually abused dozens of underage girls.

Senate Democratic Leader Chuck Schumer on Tuesday called for Acosta's resignation after accusing him of having "cut Epstein a sweetheart deal while Acosta was a US attorney in Florida in 2008."

In remarks on the Senate floor, Schumer said that "it is now impossible for anyone to have confidence in Secretary Acosta's ability to lead the Department of Labor. If he refuses to resign, President Trump should fire him."

Schumer's call comes after House Speaker Nancy Pelosi on Monday called for Acosta to resign, saying in a tweet, "(Secretary Acosta) must step down. As US Attorney, he engaged in an unconscionable agreement w/ Jeffrey Epstein kept secret from courageous, young victims preventing them from seeking justice. This was known by @POTUS when he appointed him to the cabinet. #AcostaResign."

Acosta defended his handling of the matter on Tuesday afternoon and said that he is pleased that "prosecutors are moving forward."

"The crimes committed by Epstein are horrific, and I am pleased that NY prosecutors are moving forward with a case based on new evidence," Acosta wrote in a series of tweets on Tuesday. "With the evidence available more than a decade ago, federal prosecutors insisted that Epstein go to jail, register as a sex offender and put the world on notice that he was a sexual predator."

A debate on Capitol Hill

It's not yet clear what role, if any, Congress will play in probing the deal that is now under intense scrutiny. While pressure is growing on Capitol Hill for Acosta to step down, so far the pressure is only coming from Democrats, not Republicans.

Republican Sen. Lamar Alexander, the chairman of the Health, Education and Labor Committee, said on Monday that he sees no reason for Acosta to resign.

"I see no need for that," the senator from Tennessee said when asked about the calls from some Democrats.

Alexander argued that the plea agreement Acosta oversaw when he was the US attorney in Florida was thoroughly vetted by his committee when Acosta was nominated for labor secretary and was consistent with department policy.

On Tuesday, Schumer called for hearings and said the Department of Justice should make public its review of Acosta's role in the case.

"This review cannot be kept in the dark," Schumer said, adding, "and there should be hearings."

Schumer added that President Donald Trump needs to explain his relationship with Epstein, who Trump praised in a past interview, even noting that Epstein had a fondness for young women.

Republican Sen. Lindsey Graham, the chair of the Judiciary Committee, said on Tuesday, "I think just let the professionals do it. I mean, it's not our job to be prosecutors," though he added,"if this plea deal doesn't withstand scrutiny, then it would be a job of the Judiciary Committee to find out how it got off the rails, what kinds of checks and balances do we have to make sure that complaints involving minor children are adequately investigated."

"If we think somebody is out there abusing children, no matter how hard the case may be, you want to bring it to court simply to get these people deterred, if nothing else," Graham added.

Acosta under scrutiny

Acosta's handling of the plea deal in 2008 for Epstein has come under intense scrutiny in recent months after a Miami Herald investigation.

The non-prosecution deal with federal prosecutors allowed the hedge fund manager to plead guilty to two state prostitution charges and serve just 13 months in prison.

The Herald investigation said Acosta gave Epstein the "deal of a lifetime" despite a federal investigation identifying 36 underage victims. The agreement, the Herald said, "essentially shut down an ongoing FBI probe" and further granted immunity to "any potential co-conspirators" in the case.

In February, a federal judge in Florida ruled that the Department of Justice broke the law by failing to confer with Epstein's victims about the agreement.

Acosta defended his handling of the Epstein case during his confirmation hearing in March 2017.

"At the end of the day, based on the evidence, professionals within a prosecutor's office decide that a plea -- that guarantees that someone goes to jail, that guarantees that someone register generally and that guarantees other outcomes -- is a good thing," he said at the time.

Asked in February whether President Donald Trump had lost confidence in his labor secretary, then-White House press secretary Sarah Sanders said she was "not aware of any changes in that front," and stressed the White House would be looking into the federal judge's ruling.

A senior Trump administration official was cautious this week in assessing Acosta's standing with the White House in the wake of Epstein's indictment on Monday, which alleges that between 2002 and 2005, Epstein ran a trafficking enterprise in which he paid hundreds of dollars in cash to girls as young as 14 to have sex with him at his Upper East Side home and at his estate in Palm Beach. It also alleged he worked with employees and associates to lure the girls to his residences and paid some of his victims to recruit other girls for him to abuse.

"We will wait and see what develops. This is obviously a significant event," the official said about the Epstein case. "We need to see what comes of it."

The official noted there was an internal administration review of Acosta's handling of the Epstein case. But the official could not say whether that review had been completed.

This story has been updated with additional developments Tuesday.