The culture secretary, Jeremy Hunt, has not been sacked by David Cameron because is acting as a "firewall" who is insulating the prime minister against further questions about his relationship with top News International executives, according to the Labour leader, Ed Miliband.

Speaking on Radio 4's Today programme Miliband said it "beggars belief" that Hunt is still in his job and it was "incredible" he hadn't resigned already following revelations on Tuesday that Hunt's special adviser had allegedly passed sensitive information to a senior News Corporation lobbyist with regards to an £8bn takeover bid of the broadcasting giant BSkyB.

"Why is Jeremy Hunt still in his job? Because David Cameron has questions to answer, and Jeremy Hunt is, if you like, acting as a firewall. And if he goes the questions will then move to David Cameron's conversations with Rebekah Brooks, with James Murdoch and others.

"Does this matter for politics? I say it does matter because what matters is that you govern in the interests of the British people, not in [the interests of] a few rich powerful people who have access to you."

Hunt's special adviser, Adam Smith, resigned on Wednesday after admitting he had gone "too far" in his dealings with the News Corp lobbyist Frédéric Michel, who was working for James Murdoch at the time.

"Frankly it beggars belief that he [Hunt] is still in his job because to believe he should stay you have to believe that his special adviser acted … as a lone wolf who spent six months in collusion with New Corporation, passing them information that was to be announced in the House of Commons, providing information about discussions with the regulator, providing information about what opposing parties were saying," Miliband said.

"Some special advisers have resigned in the past and their bosses haven't resigned. But that's because their bosses didn't know what they were up to. This is a completely different category."

Miliband said the £8bn takeover was "a central issue" for Hunt's department and he thought it was "incredible that Jeremy Hunt hasn't resigned and that David Cameron has kept him in his job".

Miliband also admitted getting too close to Murdoch himself. Asked earlier this morning on Radio 5 Live about attending a party thrown by the members of the Murdoch family, Ed Miliband said: "In retrospect, we all should have been more wary."

"In the end you take a judgment that obviously you want your party to do well. You want your party to get positive reporting," he admitted.

However, questioned by the presenter Evan Davis on Today about his own close relationship with the Murdochs, he said, "There is a world of difference between getting too close to the Murdochs and the kind of pattern of behaviour that we have seen revealed in the last few days at Leveson."

Miliband's comments follow an undertaking from Hunt that he is willing to hand over texts, phone records and private emails in an effort to prove he was unaware that his disgraced special adviser had acted without his authority by providing a stream of commercially confidential information to News Corporation during its bid to take full control of BSkyB.

Adam Smith fell on his sword on Wednesday an hour before prime minister's questions, admitting he had allowed the perception to develop that the Murdochs' News Corp had "too close a relationship with the culture department".

Hunt refused to resign, saying he had personally acted with scrupulous impartiality, and insisted he had been unaware of "the volume and tone" of Smith's contacts with News Corp. He said his special adviser had "overstepped the mark unintentionally".

Hunt insisted he had acted in an impartial way, pointing out that on four separate occasions he taken the independent advice of his media regulators about the bid, even though these decisions ran counter to the commercial interests of News Corp.

Although Hunt survived a torrid day, he still suffered a series of blows, leaving his political career hanging in the balance, possibly for months:

• The stockmarket watchdog, the Financial Services Authority, is understood to be investigating whether Smith's emails to News Corp amounted to a breach of market abuse rules.

• Lord Justice Leveson rejected Hunt's request to bring forward his evidence to the inquiry, meaning he will not give evidence until mid-May at the earliest, leaving open the possibility that the minister will dangle for several weeks. Leveson told the inquiry that "although I have seen requests for other inquiries and other investigations" – a coded reference to the letter he had received from Hunt – "it seems to me that the better course is to allow this inquiry to proceed" on the timetable already intended. Politicians such as Hunt, but also Cameron and Tony Blair, are scheduled to appear between mid-May and mid-June.

• Labour demanded that Cameron instruct his independent adviser on the ministerial code to investigate at least three potential breaches of the code by Hunt. The code requires ministers to be responsible for the actions of their special advisers. Ed Miliband accused Cameron of putting cronies before the country, and revived an old Labour attack on John Major's government, saying a "shadow of sleaze" hung over the government.

• Downing Street said Hunt must be willing to hand over phone records, emails and phone texts if requested by Leveson, to prove he was entirely unaware that his special adviser was criticising Ofcom and co-operating with News Corp to find ways to wave the deal through. So far, Leveson has only seen the News Corp email records of the phone calls between Hunt's office and Michel, News Corp's former public affairs director for Europe.

It has emerged that some of Michel's accounts of his contacts with government officials, including Nick Clegg's chief of staff Jonny Oates, are inaccurate or portray brief email exchanges as meetings. Tory MPs noted that Michel had in his emails repeatedly misreported his contacts with Smith as conversations with Hunt himself.

Michel has conceded that he had no direct contact with the culture secretary after he assumed responsibility for ruling on the BSkyB bid in December 2010. He insists he believed Smith was representing the views of his minister.

Number 10 conceded that Cameron had himself not conducted an inquiry into whether Smith had been truly operating independently of Hunt, saying instead the prime minister believed his culture secretary had acted properly.

Cameron rejected an inquiry by his independent adviser into the ministerial code, claiming Leveson himself had asked for no further inquiries to be launched.

Labour MPs are expected to ask Jonathan Stephens, the permanent secretary at the Department for Culture, Media and Sport, to explain his role when he appears in front of the public accounts committee. Hunt said Stevens had nominated Smith to be the point of contact for News Corp, and that this had never been seen as a secret back channel.

Smith, and potentially Hunt, could be in most trouble over one email dated 24 January 2011, in which Michel told the News Corp executive James Murdoch that he had managed to get some information on Hunt's upcoming statement to parliament, adding: "Although absolutely illegal!"

The next day Hunt announced his intention to refer the takeover bid to the Competition Commission, but only after giving News Corp more time to address concerns about "potential threats to media plurality". James Murdoch said the "illegal" reference had been a "joke" – but Labour said Murdoch got "the very words that Jeremy Hunt was going to use" before his statement to parliament.

There is no allegation that any trading on information took place. But unlike the US's security and exchange commission rules on insider trading, the UK's FSA need not show beneficial trades have been made under the market abuse rules to press charges.

Hunt's supporters, including the education secretary, Michael Gove, took to the airwaves to defend him. Rob Wilson, Hunt's parliamentary aide, drew up questions for MPs to ask after his Commons statement. In private, Conservative MPs were less positive.

One MP said the growing consensus was that Hunt would struggle to keep his job.