Assange has been holed up in the embassy since 2012, dodging a European arrest warrant and extradition to Sweden over a 2010 probe into rape and sexual assault allegations against him.
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Sweden dropped its investigation last year.
But British police are still seeking to arrest him for failing to surrender to a court after violating his bail terms during his unsuccessful battle against extradition.
Assange's legal team has asked a British court to cancel the warrant, but a judge last week dismissed his claims that the document was rendered null-and-void because there was no longer any underlying crime.
"I'm not persuaded that the warrant should be withdrawn," Judge Emma Arbuthnot told a court in London, explaining that Assange had breached his bail conditions in 2012.
But she said she would rule on Tuesday on another application from Assange's lawyers asking her to consider whether it would be in the "public interest" to keep the warrant in place.
The former hacker fears that arrest by British authorities could lead to him being extradited to the United States over WikiLeaks' publication of secret US military documents and diplomatic cables in 2010.
US Attorney General Jeff Sessions last year said his arrest was a "priority".
Assange wrote on Twitter that his latest legal argument hinged on an alleged "cover-up" by the British government to keep him detained.
'Severe punishment'
He highlighted a Guardian report which quoted emails sent by Britain's Crown Prosecution Service (CPS) to Swedish counterparts in 2012, urging them not to drop their application for a European arrest warrant.
In the emails, a CPS lawyer apparently commented on a 2012 article saying that Sweden was dropping the case by writing: "Don't you dare get cold feet!"
Swedish prosecutors told the CPS in 2013 that they "felt obliged" to lift the warrant, but only announced last year that it had finally been dropped.
The CPS also admitted it destroyed emails relating to the case after the lawyer handling the British response retired in 2014.
Ecuador's foreign ministry said it wants to reach a solution with Britain that satisfies both sides and respects Assange's human rights.
Assange's lawyer Mark Summers said his client had been living in conditions "akin to imprisonment" and his "psychological health" has deteriorated and was "in serious peril".
"The last five-and-a-half years that he has spent may be thought to be adequate, if not severe punishment, for the actions that he took," Summers told court.
The court heard that the 46-year-old was suffering from a bad tooth, a frozen shoulder and depression.
Assange only very rarely emerges onto the balcony of the embassy building, citing concerns for his personal safety, but he frequently takes part in media conferences and campaigns via video link.
Ecuador in December granted citizenship to the Australian-born Assange, and asked Britain to recognise him as a diplomat, in an unsuccessful attempt to provide him with immunity and usher him out of the embassy without the threat of arrest.
But London swiftly rejected the move.
"Ecuador knows that the way to resolve this issue is for Julian Assange to leave the embassy to face justice," the British government said.
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