The court refused to grant permission because it "did not raise an arguable point in law".
Assange (50) has been trying for years to avoid a U.S. trial on a number of charges related WikiLeaks publication of a large trove classified documents more than 10 years ago.
It is expected that the case will be sent to Priti Patel, British Home Secretary, who will decide whether or not to extradite.
An initial rejection by a British court judge of an extradition request from the United States was based on Assange's likelihood to die in harsh U.S. prison conditions. Later, U.S. authorities assured Assange that he wouldn't be subject to the extreme treatment his lawyers claimed would threaten his mental and physical health.
The High Court reaffirmed the decision of the lower court in December. It stated that the U.S. promises were sufficient to ensure Assange would be treated humanely.
Assange's options are now limited by Monday's news, but his defense team may still try to bring his case before the European Court of Human Rights. Nick Vamos, former Head of Extradition at the Crown Prosecution Service said that Assange's lawyers could also challenge other points he lost in the original district court ruling.
Barry Pollack, Assange’s U.S.-based lawyer said Monday that it was "extremely disappointing” that the British Supreme Court would not hear the appeal.
"Mr. He said that Assange would continue to fight for his extradition to the United States in order to face criminal charges of publishing truthful and newsworthy material.
Birnberg Peirce Solicitors in Britain, Assange's lawyers, stated that they could submit submissions to Home Secretary within four weeks before she makes any decision.
American prosecutors claim that Assange illegally assisted Chelsea Manning, U.S. Army intelligence analyst, to steal classified diplomatic cables. WikiLeaks published the files later on, putting lives at stake.
Lawyers and supporters of Assange claim that he was acting as journalist and therefore entitled to First Amendment freedom of speech protections for publishing documents that expose U.S. military wrongdoings in Iraq and Afghanistan. They claim that Assange's case is motivated by political motives.
Assange's lawyers claim that he could spend up to 175 years in prison in the U.S. if convicted. However, American authorities say the sentence is likely to be lower.
Since 2019, Assange is being held in Britain's Belmarsh Prison, London. He was detained for failing to pay bail during a separate legal case. To avoid being extradited to Sweden, Assange spent seven years in London's Ecuadorian Embassy.
Sweden ended its investigation into sex crime in November 2019, owing to the lack of time.