The suit named the CIA, former CIA director and former US secretary of state Pompeo, and the security firm Undercover Global and its chief executive David Morales Guillen.
Attorney General Mark Dreyfus and Foreign Minister Penny Wong responded to the British government's decision by saying Assange's "case has dragged on for too long and should be brought to a close."
UK Home Secretary Priti Patel gave the requisite ministerial sign-off for the Australian national's extradition order after several stages of court appeals right up to the Supreme Court.
The decision now rests with interior minister Priti Patel, although Assange's lawyers may still appeal to the High Court if she approves the extradition.
Assange, 50, and his South Africa-born partner Stella Moris, 38, have two sons together- four-year-old Gabriel and two-year-old Max.
The ruling brings the long-running legal saga in the UK courts closer to a conclusion unless Assange's lawyers launch a challenge on a separate point in the case.
Swedish prosecutors on Friday dropped their investigation, but Assange still faces the risk of arrest in Britain.
The woman who has accused WikiLeaks founder Julian Assange of rape has slammed the decision by Swedish prosecutors to drop an investigation as a "scandal".
Assange took refuge in Ecuador's embassy in London in 2012 to escape extradition to Sweden to answer questions about sex-crime allegations from two women.