His lawyers had argued that because Sweden ended its rape investigation against Assange, there was no longer any justification for the arrest warrant against him. The embassy is home to WikiLeaks founder Julian Assange, who is living there as he evades a warrant for his arrest on breach of bail charges.
Judge Emma Arbuthnot said in her judgement: "Many authorities underline the importance of a defendant attending court when bailed to do so and they describe the way that the administration of justice can be undermined by defendants who fail to attend".
On Tuesday, the court ruled the warrant is still valid.More news: Tillerson warns Latin America to beware of China
They said Assange's medical problems included "a awful bad tooth, frozen shoulder and depression".
His lawyer Jennifer Robinson said in a statement: "We must confront the real issue in this case - if Julian Assange walks out of the embassy today, he risks facing extradition to the U.S.to face prosecution for publishing information in the public interest with Wikileaks". Ms Espinosa said the South American country was "considering and exploring the possibility of mediation" to end the "untenable" five-year impasse after announcing it had granted Mr Assange citizenship.
The 46-year-old has been inside the Ecuadorian embassy in London in self-imposed exile since June 2012.
The 46-year-old former hacker claims he feared authorities would hand him over to the United States to face prosecution over the leak of classified military and diplomatic documents.More news: New features coming to PS4's operating system in big 5.50 update
Assange only very rarely emerges on the balcony of the embassy building, citing concerns for his personal safety, but frequently takes part in media conferences and campaigns via video link.
There are no formal charges in against Assange in the US, though news reports a year ago suggested the Justice Department was looking into it.
It is not publicly known whether there is a sealed USA indictment against Assange.
Sky's David Bowden, at the court, said the judge was now looking at case law to see if she could consider the reasonable cause argument without Assange present.More news: 'Westworld' Premieres First Season 2 Trailer During Super Bowl
She said she would make that ruling on February 13, according to The Times.
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