Beleaguered Culture Secretary Jeremy Hunt today pledged to hand over all his private texts and emails to his special adviser, to the Leveson Inquiry.
Mr Hunt said he believed that his contacts with Adam Smith would show that he handled the BSkyB takeover bid by Rupert Murdoch's News Corp empire with "total integrity".
"I will be handing over all my private texts and emails to my special adviser to the Leveson Inquiry and I am confident that they will vindicate the position that I handled the BSkyB merger process with total integrity," he told reporters outside his London home.
Mr Smith was forced to resign on Wednesday after details of his contacts with a News Corp executive were released by the Leveson Inquiry, prompting accusations of a secret "back channel" between Mr Hunt's office and News Corp.
Mr Hunt's commitment to pass material to the Leveson Inquiry is unlikely to ease pressure on David Cameron to order his own investigation into whether he breached the ministerial code of conduct.
Liberal Democrat deputy leader Simon Hughes last night became the first senior figure in his party to call for the matter to be referred to the Prime Minister's independent adviser on ministerial interests, Sir Alex Allan.
"What I cannot understand is why the matter of the ministerial code of conduct, which is to do with do you take responsibility for your special adviser, is not something the Prime Minister should immediately refer to the person who has been given the job of doing that, Sir Alex Allan," he told BBC1's Question Time.
Mr Cameron was already facing calls for Sir Alex to investigate from the Labour Party and from the senior Conservative backbencher Bernard Jenkin who chairs the Commons Public Administration Committee.
Mr Jenkin said it was "extraordinary" that a special adviser like Mr Smith had any involvement with a takeover bid in which Mr Hunt was supposed to be acting in an impartial, quasi-judicial capacity.
Mr Hunt insisted that he acted with "scrupulous fairness" in his handling of the BSkyB bid and was not in any way influenced by Mr Smith's contacts with News Corp executive Frederic Michel.
Labour, however, say that under the code of conduct, he must take responsibility for his special adviser's actions.
Labour leader Ed Miliband said Mr Cameron's refusal to order an inquiry suggested he was trying to protect his own position.
"Every day David Cameron looks more like a Prime Minister organising a cover-up rather than standing up for the public," he said.
"First he refuses to sack Jeremy Hunt despite the weight of evidence against him. Now despite all-party calls to do so, he refuses even to ask the independent adviser on ministerial interests to examine whether Mr Hunt broke the ministerial code.
"People watching this Prime Minister's actions will conclude he has a great deal to hide and that his real fear is that Jeremy Hunt's resignation will put his own actions and behaviour centre-stage."
Labour deputy leader Harriet Harman said that Mr Hunt should release the texts and emails to Parliament and not just the Leveson Inquiry.
"Effectively what we have got here, which is very disappointing, is the Government using Lord Leveson's inquiry as a big carpet under which they can sweep everything," she told BBC News.
"Lord Leveson's inquiry is enormously important but it doesn't relieve the Prime Minister of his duties as prime minister. It doesn't relieve secretaries of state of their obligations to live up to the ministerial code."
Asked whether Mr Cameron agreed with Mr Hughes that an inquiry into potential breaches of the ministerial code was needed, the Prime Minister's official spokesman said: "No. The Prime Minister made very clear on Wednesday that he has full confidence in the way Jeremy Hunt handled the BSkyB bid.
"Jeremy Hunt made clear in his statement to the House of Commons that at all times he took the advice of independent regulators and the permanent secretary.
"Clearly, matters relating to the code are matters for the Prime Minister. However, there is a public inquiry under way and we believe it is right to allow that inquiry to run its course."
Responding to the announcement that Mr Hunt plans to hand over emails and text messages relating to the BSkyB bid to Lord Justice Leveson, the spokesman said it was for the Culture Secretary to prepare and present his own submission. There was no "central clearance process" for ministers to clear their evidence to the inquiry with Number 10.
Asked about Mr Miliband's claim of a cover-up, the spokesman said: "There is a public inquiry. That public inquiry is looking at issues relating to the BSkyB bid. That public inquiry has the full force of the law as set out by the 2005 Public Inquiries Act and is being held in public.
"I am not sure that is consistent with what you say."
Conservative Party deputy chairman Michael Fallon took to the airwaves to defend Mr Hunt, saying it was "absurd" for Ms Harman to suggest the Culture Secretary was guilty before he had even given evidence to the Leveson Inquiry.
Responding to Mr Hughes's call for the case to be referred to the independent adviser on the ministerial code, Mr Fallon told Sky News: "He's entitled to his views, but the point really here is that the person heading the inquiry - Lord Justice Leveson - has himself said that the better course is to allow his inquiry to proceed.
"Jeremy Hunt will appear in front of it. It won't be months away - he has asked for his appearance to be brought forward. He has said today he's going to publish all the texts and emails for the inquiry. He will be tested on that evidence.
"During that time, he remains accountable to Parliament and Harriet Harman can ask him any question she likes. But the better course is to allow this proper judicial inquiry that is under way to proceed."