The party said the process would have "more credibility" if Deputy First Minister Nicola Sturgeon or a senior member of the Government took charge instead of Alex Salmond.
The call came after Lord Justice Leveson found that Mr Salmond would have knowingly led UK ministers to break the law if it advanced Scottish interests.
The First Minister displayed a "striking" readiness to lobby UK Business Secretary Vince Cable and former culture secretary Jeremy Hunt on behalf of Rupert Murdoch's NewsCorp during their consideration of the legality of its proposed acquisition of BSkyB, according to the Inquiry Report released yesterday.
However, Lord Justice Leveson found no evidence of a deal to trade NewsCorp newspaper support for the SNP in exchange for Scottish Government support for the BSkyB bid, but he noted that both were discussed during the same conversation.
Ultimately, Mr Salmond did not contact the UK ministers, despite indicating his willingness to do so, and therefore "cannot be criticised" because he must be "judged by what he did, as opposed to what he said he was prepared to do".
Earlier this week, Mr Salmond proposed that an independent group should look at how best to implement the findings of the Leveson report north of the border.
The First Minister said he wanted to achieve cross-party agreement on what needed to be done in the wake of it.
Responding to the report's findings today, Labour MSP Patricia Ferguson, the party's culture spokeswoman, said that Mr Salmond had displayed "flawed" judgment in his dealings with NewsCorp.
She told BBC Radio Scotland: "This is entirely about getting the system right for the victims and it's about making sure that system has the maximum amount of credibility.
"We think that Mr Salmond's judgment in this entire scenario has been flawed and we think therefore that he should stand aside from this and allow either his deputy of another senior member of his government to take over the task."
Ms Ferguson denied suggestions she was seeking to score political points.
She told the Good Morning Scotland programme: "It's very much not about that. We want to see press regulation that is fair, that is just and that works well in Scotland.
"Mr Salmond has shown that his judgment on this issue is not what it should be. He should stand aside.
"It isn't that we're saying his proposals are all wrong. We want to hear what his proposals are and we want to work with the Government to get in place a system that is appropriate for Scotland, taking on board the Leveson points.
"But I think it is very interesting that Leveson has devoted so much time, an entire section of his report, to his comments about Mr Salmond."
However, Culture Secretary Fiona Hyslop told the BBC that Mr Salmond should "of course" be involved in the post-Leveson process.
She said: "It's really important that the opposition parties rise to their responsibilities, put their petty party politicking to one side and remember that quite clearly in the report, Lord Leveson said that Alex Salmond could not be criticised and recognised that his motivation was for jobs for Scotland."
Lord Justice Leveson found that, in respect of the BSkyB takeover issue, Mr Salmond was "motivated by an anxiety to help Scottish employment and to benefit Scotland generally".
Ms Hyslop said there was a chance to build a cross-party consensus on the way forward in light of the Leveson report.
She said: "I think the implementation group that the First Minister is establishing is the right way forward, but it has to be done in consensus.
"I think we could lead the way on that and I really would ask the opposition parties to move on from their personal dislike of Alex Salmond and actually do what the public expects, and get on with the job."
The Culture Secretary said she rejected state regulation of the press.
She said: "The point in the middle where I think Leveson is pointing to is independent regulation that has some kind of statutory regulation, but certainly not setting out rules for the press. The state should never do that."
Scottish Secretary Michael Moore said two principles had to be balanced when choosing a way forward in the wake of the Leveson report.
The Liberal Democrat MP told Good Morning Scotland: "We believe that it is fundamentally important that we have a balance of principles here, and they are very solid principles, that we on the one hand have a free raucous press which is able to hold politician, business, whoever, to account.
"That's something that's very precious to the United Kingdom and we should seek to maintain that.
"However, that same raucous press can't have no restraint whatsoever in terms of being able to bully vulnerable people.
"I think the moment that crystallised all of this for everybody in the country was to discover that the parents of Milly Dowler had had phones hacked.
"We saw a whole new aspect to the behaviour and contempt of some of the press for individuals.
"Politicians and celebrities aside, I think people then realised there was a really serious issue.
"What I think Lord (Justice) Leveson has pretty skilfully done is seek to balance these two principles - that you should protect the vulnerable while maintaining a free press - by proposing a new form of self regulation for the press which is then backed up by some legislation which simply ensures that we don't have the situation at the moment where, if people opt out of that self regulation, there's no punishment, no recourse."
The Scottish Conservatives and Liberal Democrats have previously called for Mr Salmond to step back from the proposed all-party talks on the future of press regulation north of the border.
Tory leader Ruth Davidson said the Leveson report contains "some marked criticisms" of the First Minister and his conduct surrounding the BSkyB bid.
"He clearly still has questions to answer," she said.
"In light of these remarks, I think it would be wise for the First Minister to take a step back from the process and allow another minister to represent the Scottish Government in the proposed cross-party talks."
Willie Rennie, leader of the Scottish Liberal Democrats, said: "It is not clear to me how Mr Salmond is going to be able to convene a cross-party group of leaders to take forward the report given he has been so heavily criticised in the report.
"He should offer to stand aside from the cross-party work. It should be led by someone untainted by the report."
An SNP spokesman said that any refusal by the other parties to "engage fully and constructively with the cross-party talks the First Minister has proposed would look silly and petulant - and would go back on the unequivocal undertakings they gave to Parliament only yesterday".
He stressed: "This is a serious issue and should be treated as such. That means a grown-up, consensual approach is needed - not petty point-scoring."
