The judges said they wanted the full details of the alleged torture to be published in the interests of safeguarding the rule of law, free speech and democratic accountability.
But they had been persuaded that it was not in the public interest to publish those details as the US government could then "inflict on the citizens of the United Kingdom a very considerable increase in the dangers they face at a time when a serious terrorist threat still pertains".
These details were a summary of a report by the US government to the British security services about the detention and treatment of the individual concerned which amounted to 25 lines.
No 10 said it was not aware of any threat from the US government to withdraw intelligence co-operation with Britain if details of the case were revealed.
The allegations relate to Binyam Mohamed, a British resident held in Guantanamo, who alleges he was tortured in Pakistan, Morocco and Afghanistan between 2002 and 2004.
Mr Davis said a High Court ruling, which pointed to complicity by the UK and US authorities in his torture, was prevented from being published after the US put pressure on the UK.
The former shadow home secretary raised the case in a point of order in the House of Commons.
He said Mr Miliband should make a statement to MPs about the issue as soon as possible to "explain what the devil is going on".
He said the UK government should make it "plain" that it did not support torture in any circumstances.
Mr Mohamed, 30, has been held in Guantanamo for four years after the US authorities accused him of conspiring with Al-Qaeda leaders to plan a series of terrorist attacks.
But war crimes charges against him were dropped in October.
Mr Mohamed - an Ethiopian national who moved to the UK in 1994 - says documents support the fact that the evidence against him was obtained through torture.
Last August, Lord Justice Thomas said evidence relating to the case should be disclosed, saying it was "essential".
However, the British government argued the disclosure of certain material would cause "significant damage to national security".
'None of their business'
Mr Davis said it appeared the Bush administration had "threatened" the UK government about the repercussions should details of the case be made public.
"Frankly it is none of their business what our courts do," he said, adding this was "plain fact" not merely an allegation.
"They should not seek in any circumstances to put pressure on British courts. That's completely beyond the rule of law."
There is no other terms for what the US intelligence services are doing than blackmail
Nick Clegg, Lib Dem leader
He said Mr Miliband must explain why this had happened and whether the new Obama administration supported its predecessor's stance on the issue.
"While he is at it, he [the foreign secretary] should explain what degree of complicity we have in this," he told the BBC.
Mr Davis said the government had taken a "highly principled public stand" against torture but must "come clean" about whether there were cases where British agencies ever knew about instances of torture by others.
The BBC's Jonathan Beale in Washington said a former Bush administration official who dealt with Guantanamo Bay confirmed that US intelligence agencies did tell the UK that they opposed the release of certain US intelligence without their consent.
The official added that this was standard practice, just as MI6 and MI5 would oppose the release of UK intelligence in the US without their consent.
Civil liberties campaigners described the judges' remarks on the case as "astounding".
Shami Chakrabarti, director of Liberty, said the Bush administration had tried "to bully" the British courts and President Obama must make it clear he would not do the same.
And Lib Dem leader Nick Clegg said all the documents in the case must be published immediately.
"There is no other terms for what the US intelligence services are doing than blackmail," he said.
"It is simply incredible that the US government would have halted intelligence co-operation with the UK if this information had been made public."
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