The Labour leader, Ed Miliband, has called for the resignation of the culture secretary, Jeremy Hunt, over allegations that he set up a private back channel to Rupert Murdoch's News Corporation at a time when he was charged with making a quasi-judicial decision on whether to allow its takeover of BSkyB.

Miliband said the Conservative MP Hunt was "standing up for the interests of the Murdochs" rather than those of the British people, after it was alleged in evidence to the Leveson inquiry that he had provided detailed information to James Murdoch on the state of the bid, as well as the thinking of the media regulator Ofcom.

"He should resign. He himself said that his duty was to be transparent, impartial and fair in the BSkyB takeover. But now we know that he was providing advice, guidance and privileged access to News Corporation. He was acting as a back channel for the Murdochs," Miliband said in a statement.

"He cannot stay in his post. And if he refuses to resign, the prime minister must show some leadership and fire him."

Hunt said in a statement on Tuesday night that he had asked Lord Justice Leveson to bring forward his appearance before the inquiry and was "confident that when I present my evidence the public will see that I conducted this process with scrupulous fairness". He said: "We've heard one side of the story today but some of the evidence reported meetings and conversations that simply didn't happen."

Harriet Harman, Labour's shadow culture secretary, had earlier called for Hunt's resignation. She told the Commons the right thing would be for the culture secretary to come to the chamber to apologise and to resign.

Chris Bryant, a former Labour shadow culture minister who was a victim of News of the World phone hacking, also suggested issues of privilege might arise, including whether a minister had misled the house.

There were also calls for the Scottish secretary, Michael Moore, to come to the chamber to address allegations made at the Leveson inquiry on Tuesday concerning the conduct of the Scottish first minister, Alex Salmond, during News Corp's Sky takeover bid.

During the appearance by James Murdoch, News Corp's deputy chief operating officer, before Lord Justice Leveson on Tuesday, the inquiry was shown extracts from 163 pages of email correspondence from Fréd Michel, News Corp's director of public affairs for Europe during the Sky takeover bid, to Murdoch marked "confidential", detailing communications with Hunt, often via Hunt's special adviser, Adam Smith.

Hunt's aides also stood by Smith, and suggested Michel had wildly exaggerated the level of his knowledge and contacts with Hunt's Department for Culture, Media and Sport (DCMS) during the decision-making process between June 2010 and July 2011.

David Cameron's spokesman insisted the prime minister had confidence in the culture secretary. He also said he would not provide a running commentary on a judicial process, referring to the Leveson inquiry.

The inquiry heard on Tuesday afternoon that Hunt's office had asked News Corp for "input" into a Commons statement made by the culture secretary on the bid for BSkyB in January 2011.

One email sent by Michel on 23 January 2011 drew gasps in court 73 as the counsel for the Leveson inquiry, Robert Jay, read out the lobbyist's boasts about how he had obtained information from Hunt about a statement he was about to make on the BSkyB bid: "Confidential. JH statement. Managed to get some infos on the plans for tomorrow (although absolutely illegal!)" Murdoch told the inquiry he thought this was a joke.

Hunt's DCMS officials, speaking before they had had a chance to look at the emails handed to Leveson in detail, said the culture secretary had not spoken to Michel on a daily basis and was willing to make his phone records available to check the veracity of the claims made at the inquiry.

His officials said Michel had exaggerated his knowledge of what the government was thinking and was trying to present his employers with an impression that he had an inside track.

Hunt's aides said the culture secretary had always been minded to send the Sky takeover to the Competition Commission, adding that the bar for giving it the go-ahead had been raised by the behaviour of the business secretary, Vince Cable.

Cable was forced to hand over ministerial responsibility for the bid to Hunt in December 2010 after telling undercover Daily Telegraph reporters that he was at war with Rupert Murdoch, the News Corp chairman and chief executive. DCMS officials said they were not aware whether Hunt had spoken directly to the prime minister about the allegations.

News Corp withdrew its bid to buy the 61% of Sky it did not own in July 2011 after it became engulfed in the scandal surrounding the hacking of the murdered schoolgirl Milly Dowler's phone by the News of the World.

The Michel emails published by the Leveson inquiry reveal that sometimes half a dozen confidential texts and emails a day would fly back and forth between the culture secretary's Cockspur Street office just off Trafalgar Square in central London and the News Corp team promoting the Sky takeover bid.

Murdoch, in his evidence to the inquiry, steadfastly defended Michel's relationship with Hunt's office, saying it was "active public affairs engagement" and "legitimate advocacy".

On the eve of one key government announcement in March 2011, Michel emailed his boss excitedly at 3am: "Urgent. JH decision … He is minded to accept … and will release around 7.30am to the market." Michel wrote of Hunt: "He said we would get there in the end and he shared our objectives."

On 23 January 2011, before Hunt was due to make a Commons statement about the Sky bid, Michel told Murdoch: "He [Hunt] still wants to stick to the following plan … His view is that once he announces publicly he has a strong UIL [undertaking in lieu, to spin off Sky News to deal with issues over plurality], it's almost game over for the opposition … He very specifically said he was keen to get to the same outcome and wanted JRM [James Murdoch] to understand he needs to build some political cover on the process."

The following day, an email sent at 3.21pm shows Murdoch being supplied with the wording of Hunt's crucial, and market-sensitive, official statement, due to be delivered the next day.

"Confidential: Managed to get some infos on the plans for tomorrow (although absolutely illegal!). Press statement at 7.30am … Lots of legal issues around the statement so he has tried to get a version which helps us … JH will announce … that he wishes to look at any undertakings that have the potential to prevent the potential threats of media plurality."

Michel has told the Leveson inquiry that his messages that habitually referred to conversations with Hunt were, in truth, conversations with Hunt's staff, often his special adviser Adam Smith.