Acosta was confirmed to President Trump's Cabinet in 2017 with a bipartisan vote of 60-38.
Acosta was not Trump's first choice to fill the post as he became president.
His resignation extends record turnover at the highest levels of Trump's administration, with acting secretaries at key departments, including the Pentagon and Homeland Security.
Mr Acosta was the U.S. attorney in Miami when he oversaw a 2008 non-prosecution agreement with Epstein.
The federal prosecutors in NY said they were not bound by the deal arranged by Acosta, which allowed Epstein to plead to a lesser offence and serve 13 months in jail with leave during the day while registering as a sex offender.
On Wednesday Acosta held a news conference to defend his handling of the deal, which allowed Epstein to plead guilty to a state charge and not face federal prosecution. He paid settlements to victims and is a registered sex offender.
The globe-trotting former hedge-fund manager - who was arrested in New Jersey last week for allegedly trafficking underage girls to sexually abuse them - doled out a total of at least $350,000 in bribes to two people previous year, prosecutors say.
Acosta's work aligned with the President's priorities while serving as secretary by reducing unemployment, as well as implementing the President's executive order that pushed private sector investment toward apprenticeships and work training. In a 50-plus-minute lawyerly rebuttal, Acosta argued his office had secured the best deal it could at the time and had worked in the best interests of Epstein's victims.
She spoke hours after Trump hailed Acosta as a "great" labor secretary, and expressed sympathy for Acosta's predicament of being questioned about the deal he reached, comparing it to his own struggles negotiating deals with Democrats. "We believe that we proceeded appropriately".
Pressed on whether he had any regrets, Mr Acosta repeatedly suggested that circumstances had changed since then.
Asked if he would make the same deal now, Acosta responded: "We live in a very different world".
Epstein, who has ties to Trump and former President Bill Clinton, was charged in a Manhattan federal court on Monday on sex-trafficking charges. He has always been surrounded by the rich and powerful, including President Trump, former President Bill Clinton and the UK's Prince Andrew. "It is even said that he likes handsome women as much as I do, and many of them are on the younger side", he told New York Magazine. At the time, the then-New York real estate mogul called "Jeff" a "terrific guy". "It is even said that he likes handsome women as much as I do, and many of them are on the younger side".
"If they let him out my guess is he'll never return", Kluvin says. Acosta had explained, breezily, apparently, that back in the day he'd had just one meeting on the Epstein's attorneys because he had "been told" to back off, that Epstein was above his pay grade. If Trump nominates a permanent replacement, the nominee would have to go through a Senate confirmation process.