Mr Assange, 48, spent much of the second day of the hearing in the Woolwich Crown Court propped on an elbow looking at a skylight.
In the afternoon he appeared to be drifting off, prompting the judge to direct a member of his legal team, Gareth Pierce, to check on him.
"Mr Assange is struggling, he's finding it hard to concentrate, he can't communicate with his legal team and he's finding it very difficult," Mr Pierce told the judge.
Judge Vanessa Baraitser said she did not have the legal power to comment or rule on Mr Assange's conditions, but encouraged his legal team to formally bring the matter up with the prison.
"I think everyone in the court supports a fair hearing," she said.
Earlier, lawyers for the US government said Mr Assange published the real names of hundreds of US informants, risking their safety, in classified diplomatic cables in September 2011.
"Mr Assange did not have to publish the unredacted cables, but he decided to do so," James Lewis QC said.
But barrister Mark Summers, for Mr Assange, said the files and the password protecting them were already in the public domain, including on content-sharing website PirateBay.
The password was also the name of a chapter in Guardian journalist David Leigh's book about Mr Assange, published earlier in 2011.
"Far from being a reckless, unredacted release ... what actually occurred was that one of the media partners published a book in February 2011 and published the password to the unredacted materials, which then enabled the entire world to publish those unredacted materials in a book and they circulated on the internet, not on the WikiLeaks site, but on other sites," Mr Summers said.
"None of them have been prosecuted, some of which are US-based, all of them published first, some of them are still there."
The Guardian newspaper said Leigh had been told by Mr Assange the password was temporary and his book didn't say where the digital files were.
"No concerns were expressed by Assange or WikiLeaks about security being compromised when the book was published in February 2011," The Guardian said in a statement on Tuesday.
Mr Summers said for months WikiLeaks had carefully worked to redact the files with media partners including The Guardian, Der Speigel, Le Monde, El Pais and the New York Times.
The US State Department had even been "feeding suggested redactions" to the media partners.
Der Speigel investigative journalist John Goetz described WikiLeaks' redactions as "extreme" at the time, Mr Summers said.
Mr Summers did not explain why Mr Assange decided to go ahead and publish the unredacted diplomatic cables in late 2011.
The US government is trying to extradite Mr Assange to face 17 charges of violating the US Espionage Act and one of conspiring to commit computer intrusion over leaking and publishing thousands of classified US diplomatic and military files.
The charges carry a total of 175 years' imprisonment.