"No appeal to the High Court has yet been filed by him in respect of the other important issues he raised previously," his lawyers Birnberg Peirce Solicitors said in a statement last month.
"That separate process of appeal has, of course, yet to be initiated."
The case has become a cause celebre for media freedom, with Assange's supporters accusing Washington of trying to muzzle reporting of legitimate security concerns.
Outside the court, protesters held aloft placards imploring "Don't extradite Assange" and attached yellow #FreeAssange ribbons to walls outside the court.
Journalist and Assange campaigner Carolina Graterol, 55, from Venezuela, called the ruling "another day in the death of democracy and the rule of law in the United Kingdom."
"The position that the judge has taken in signing the extradition request... is a travesty of justice," she told AFP.
"I think he will die sadly in jail if he is sent."
Assange was last month denied permission to appeal to the UK Supreme Court against moves to extradite him to the US, where he could face a lifetime in prison.
Washington wants to put him on trial in connection with the publication of 500,000 secret military files relating to the US-led wars in Iraq and Afghanistan.
In January last year, the 50-year-old Australian appeared to have won a reprieve on the grounds he was a suicide risk if he was kept in solitary confinement at a maximum security US facility.
But the US government appealed, and at a two-day appeal hearing in October its lawyers pointed to diplomatic assurances that Assange would not be held in punishing isolation at a federal supermax prison, and would receive appropriate care.
Assange appealed that ruling and, in January, two judges allowed him to apply to the country's highest court on "points of law of general public importance".
But the court refused permission to appeal, saying the application "didn't raise an arguable point of law".
Magistrate Paul Goldspring on Wednesday said he was "duty bound" to send the case to Patel in the light of the higher court's ruling.
Assange is wanted to face trial for violating the US Espionage Act by publishing military and diplomatic files in 2010.
He could face up to 175 years in jail if found guilty, although the exact sentence is difficult to estimate.
He has been held on remand at a top-security jail in southeast London since 2019 for jumping bail in a previous case accusing him of sexual assault in Sweden.
That case was dropped but he was not released from prison after serving time for breaching bail on the grounds he was a flight risk in the US extradition case.
Assange, who married his fiancee Stella Moris in jail last month, spent seven years at Ecuador's embassy in London to avoid being removed to Sweden.
He was arrested when the government changed in Quito and his diplomatic protection was removed.
Moris last month called on Patel to block the extradition, saying: "This is a political case and she can end it."