The Liberal Democrats were thrown into fresh crisis on Saturday as their only MP in Scotland, Alistair Carmichael, faced the prospect of a full parliamentary inquiry and a byelection fight against the SNP over his role in a Whitehall leak intended to damage Scotland’s first minister, Nicola Sturgeon.

The future of Carmichael, who served as Scottish secretary in the coalition government, is being called increasingly into question by MPs and peers after the SNP triggered a formal process that could lead to his suspension from the House of Commons and a new election in his marginal Orkney and Shetland seat.

After the Lib Dems suffered the devastating loss of all but eight of their 56 MPs at the election, the SNP believes it can now inflict further humiliation by wiping them out north of the border, reducing them to seven members in the Commons.

Carmichael held on by just 817 votes, despite a strong charge by the second-placed nationalists, who cut his majority from more than 10,000 at the 2010 election.

He admitted on Friday – following a leak inquiry by the cabinet secretary Sir Jeremy Heywood – that he had authorised his special adviser to make details of the memo known to the Daily Telegraph, which ran the story at the start of the election campaign.

The memo, written by a Foreign Office official, suggested that Sturgeon had told the French ambassador in February, during a private conversation, that she wanted David Cameron to become prime minister. Both Sturgeon and the ambassador denied that she had said anything of the sort, though the claim was embarrassing to the SNP leader, whose public line was that she wanted to work with Labour to keep the Tories out of power.

Carmichael, who appeared to be losing the support of some key constituents yesterday, said he had been guilty of an “error of judgment” and acknowledged that if he still been a minister it would have been a resigning matter. He and his adviser, Euan Roddin, said they would not take severance payments.

The SNP’s deputy leader, Stewart Hosie, turned up the pressure on Saturday by calling for Carmichael to step down and demanded a full investigation by the parliamentary commissioner for standards, Kathryn Hudson.

Hosie said: “Mr Carmichael has no credibility in continuing as an MP and in my opinion he should stand down. It is appropriate for the standards commissioner to get involved and arrive at her own conclusions, having investigated the full facts.

“This dirty trick has all the appearance of being orchestrated when Alistair Carmichael was an MP, before the election campaign. Mr Carmichael went on TV after this bogus story broke, denying that he had anything to do with the dirty trick. He said that he knew nothing about it until he got a call from a journalist. We now know all of this to be 100% untrue.

“The dirty trick was bad enough – but his blatant attempt to cover it up until after the election is 10 times worse.”

On 5 April, Carmichael flatly denied having had any role in the leak. He told Channel 4 News: “I’ve told you the first I became aware of this, and this is already on the public record, was when I received a phone call on Friday afternoon from a journalist making me aware of it.”

Alistair Carmichael apologises for anti-Sturgeon leak – video Guardian

If the commissioner for standards believes there is reason to investigate, which seems likely, she will submit her findings to the parliamentary committee on standards, which has the power to recommend sanctions, including the suspension of an MP. The recommendation is then voted on by MPs on the floor of the Commons.

If MPs were to vote to suspend Carmichael for 10 days or more, this would then automatically trigger a new process under the Recall of MPs Act (championed by Nick Clegg): if at least 10% of Carmichael’s constituents sign a petition calling for a byelection, then one must be held. Carmichael could stand again but would face a huge struggle to see off the rampant SNP. Some believe Carmichael might try to argue that the investigation should not take place as he was not an MP when the Daily Telegraph published the story on 3 April, as parliament had concluded its five-year term on 30 March. However the cabinet secretary’s inquiries revealed phone calls between Roddin and a Telegraph journalist had taken place on days before publication.

Labour peer Dale Campbell-Savours, a former member of the Commons standards and privileges committee, said he believed such a defence would be unsustainable and predicted that there would be a byelection.

“It is inconceivable that a parliamentary committee could issue a penalty in a case of this gravity of less than 10 days’ suspension. Therefore a byelection will inevitably be triggered.”

He added: “There is no greater sin that an MP can commit than to lie to his electorate.”

The Lib Dems said on Saturday that Carmichael would not face any disciplinary proceedings inside the party. But the SNP also wants to know what other senior Lib Dems, including former leader Clegg and Scottish leader Willie Rennie, knew about the leak.

Hosie said: “There are also questions for Scottish Lib Dem leader Willie Rennie and indeed former Lib Dem UK leader Nick Clegg to answer – what did they know, and when did they know it?”

Billy Fox, the deputy leader of Shetland Islands Council, said: “I can’t speak for the council but my personal view is that his position to stay as an MP is untenable. His credibility is completely shot.”

Another councillor from the Shetland area added: “I think he’s had it.”