The First Minister came under fire last week over the suggestion he would lobby Culture Secretary Jeremy Hunt on behalf of the Murdochs.
Today Mr Salmond - who met Rupert Murdoch as recently as February - came under attack again, as opposition leaders in Scotland said the report by the Commons Culture, Media and Sport Committee raised questions about his judgment.
The committee's report into the News of the World phone hacking scandal accused the News Corp chief of exhibiting "wilful blindness" towards the wrongdoing in his organisation.
In light of that, Scottish Labour leader Johann Lamont demanded: "What we need to know is why Alex Salmond thinks that Rupert Murdoch is still a fit and proper person to run media in this country."
Meanwhile, Scottish Liberal Democrat leader Willie Rennie claimed the "damning" report by the Commons committee "further calls into question the First Minister's judgment".
At the same time MPs on the committee called for Mr Salmond to mount inquiries into both phone hacking in Scotland and the treatment of former MSP Tommy Sheridan who was jailed for perjury last year.
Labour's Tom Watson demanded an investigation into phone hacking north of the border in the wake of revelations at the weekend that former First Minister Jack McConnell and his family may also have been victims.
Lord McConnell said he was "speaking to solicitors" after police said his and his two children's telephone numbers were found in the notes of private detective Glenn Mulcaire who worked for News International.
Mr Watson said: "Now that we know that the former first minister in Scotland was also a target of hacking, I am writing to Alex Salmond to recommend he sets up an inquiry by the Scottish Parliament into how and why MSPs were targeted."
The call came as an aide to the First Minister revealed her phone may also have been hacked.
South of Scotland SNP MSP Joan McAlpine, writing in the Daily Record newspaper, said detectives found her mobile and home phone numbers, as well as address, in notebooks belonging to Mr Mulcaire.
Mr Watson also raised fears that the guilty verdict against Tommy Sheridan may be "unsound".
The ex-MSP was jailed for perjury after he was found guilty of lying under oath during his successful defamation action against the News of the World in 2006.
But Mr Watson said the jury in the case may not have had all the facts and urged Mr Murdoch to "order an urgent review of the information his company provided to the jury in the Sheridan case".
Meanwhile, Labour MP Jim Sheridan said Mr Salmond should consider launching an inquiry into the way Tommy Sheridan - no relation - was treated.
The MP, a member of the Culture Committee, said: "There is some suspicion surrounding the situation of the First Minister Alex Salmond and his dealings with Rupert Murdoch, hence the reason why the Murdoch newspapers have swung behind and supported the SNP in Scotland. That begs the question: why?
"Even at this late stage, I would ask Alex Salmond to come clean, look at this report, learn from this report and, more importantly, act on this report. Someone lost his liberty as a result of the evidence given and he and his family deserve some kind of inquiry into why that happened."
In its report, the Commons Culture Committee said News Corp was guilty of "huge failings of corporate governance" and that throughout its instinct had been "to cover up rather than seek out wrongdoing and discipline the perpetrators".
It said Rupert Murdoch's son James had demonstrated "wilful ignorance" about what had been going on.
However, the most damning judgment was reserved for Rupert Murdoch.
"On the basis of the facts and evidence before the committee, we conclude that, if at all relevant times Rupert Murdoch did not take steps to become fully informed about phone hacking, he turned a blind eye and exhibited wilful blindness to what was going on in his companies and publications," the report said.
"This culture, we consider, permeated from the top throughout the organisation and speaks volumes about the lack of effective corporate governance at News Corporation and News International.
"We conclude, therefore, that Rupert Murdoch is not a fit person to exercise the stewardship of a major international company."
Afterwards Mr Rennie blasted: "This is damning report which further calls into question the First Minister's judgment."
The Lib Dem MSP added: "The committee has concluded that Mr Murdoch is not a fit person to run a major media organisation. Anyone who has followed the phone hacking of Milly Dowler, 7/7 victims and families of fallen soldiers in Afghanistan will have reached that conclusion long ago.
"The fact remains that Mr Salmond thought it was appropriate to court the troubled media mogul even after these revelations."
He demanded: "Mr Salmond must explain why he thought Mr Murdoch was a fit person for the First Minister of Scotland to meet."
Meanwhile, Ms Lamont said: "Only Alex Salmond and the SNP think that Rupert Murdoch is a fit and proper person to run newspapers and television companies in this country."
She added: "We know that the First Minister lobbied on Murdoch's behalf, having first denied it. But Alex Salmond finally admitted it."
The Labour leader added: "What we need to know is why Alex Salmond thinks that Rupert Murdoch is still a fit and proper person to run media in this country. We need to know why AlexSalmond thinks it is fit and proper for Scotland's First Minister to lobby on his behalf.
"What we need is for the First Minister to explain himself to the Scottish Parliament at the first opportunity."
Last night it emerged that Mr Salmond's office had made repeated requests to set up a telephone conversation between the First Minister and Mr Hunt to discuss News Corp's bid to take over the broadcaster BSkyB.
It was suggested during the Leveson inquiry into press standards last week that Scotland's First Minister may have been prepared to intervene on behalf of Rupert Murdoch and lobby Mr Hunt.
In an email sent by Frederic Michel, director of public affairs for Europe at News Corps, to James Murdoch, it was suggested Mr Salmond would call the Culture Secretary "whenever we need him to".
Mr Salmond denies any wrongdoing, describing the suggestion he would contact Mr Hunt as "email tittle-tattle".
Earlier it emerged Mr Salmond's office had contacted the Culture Secretary's team on March 3 last year to arrange a telephone conversation between the two men
And last night Mr Hunt claimed that Mr Salmond's office had made repeated requests for a telephone conversation following the March 3 call, despite the Culture Secretary having already taken up the quasi-judicial role in deciding whether the bid for BSkyB should be referred to the Competition Commission.
In a written parliamentary answer, Mr Hunt said: "Mr Salmond's office contacted mine to request a telephone call on March 3 2011, and again a number of times in the following days. However, I can confirm that no such call took place or was ever scheduled."
The Culture Secretary added: "I can confirm that I did not at any time discuss the BSkyB bid with Alex Salmond."
A Scottish Government spokesman said last night that Mr Hunt's answers "confirms exactly what we have been saying about this issue: no call was scheduled and none took place".
He added: "The First Minister was prepared to make such a call to Mr Hunt in the interest of jobs and investment in Scotland but the opportunity never arose as things were overtaken by events."
PA
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