Sir Keir said the issue was not one of individuals but of principles.
“If you’re going to have integrity in the Westminster Parliament or the Scottish Parliament, breaches of the ministerial code in either parliament ought to lead to a resignation,” he said.
Scotland’s first minister is currently awaiting the findings of a report into whether or not she breached the code.
Lawyer James Hamilton is looking into whether the SNP leader lied to the Scottish Parliament over her knowledge of allegations against her predecessor Alex Salmond.
A separate inquiry by a Holyrood committee has concluded it is “hard to believe” she was unaware of concerns about the former first minister’s behaviour before November 2017, according to reports.
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New claims also suggest members of the inquiry have concluded that Ms Sturgeon misled their Holyrood committee if she did have knowledge of concerns, relating to the then-First Minister’s alleged behaviour towards female Edinburgh Airport staff, before November 2017.
The committee also believes she should have informed her Permanent Secretary, Scotland’s most senior civil servant, as soon as possible after an April 2018 meeting with Mr Salmond, according to Sky News.
Ms Sturgeon described reports that the committee has concluded she misled them as a “very partisan leak” from the inquiry.
On Thursday evening, she told Sky News: “I stand by all of the evidence I gave to the committee, all eight hours’ worth of evidence.
“What’s been clear is that opposition members of this committee made their minds up about me before I uttered a single word of evidence, their public comments have made that clear.
“So this leak from the committee – very partisan leak – tonight before they’ve finalised the report is not that surprising.”
A Scottish Parliament spokeswoman said the committee, which is made up of four SNP members and five from other parties, is still considering its report.
It is expected to be published on Tuesday.
The Committee on the Scottish Government Handling of Harassment Complaints was set up after Mr Salmond was awarded a £512,250 payout over the Scottish Government’s investigation into the allegations.
A spokesman for the First Minister said: “The First Minister told the truth to the committee in eight hours of evidence, and stands by that evidence.
“It is clear from past public statements that opposition members of this committee had prejudged the First Minister at the outset of the inquiry and before hearing a word of her evidence, so this partisan and selective briefing -before the committee has actually published its final report – is hardly surprising.
“The question of the First Minister’s adherence to the ministerial code is being considered independently by James Hamilton, and we expect to receive and publish his report soon.”
Scottish Conservative leader Douglas Ross has called for Ms Sturgeon to resign and said his party was “really only waiting for confirmation of what we already know.
“We have detailed that the First Minister misled the Scottish Parliament. Nicola Sturgeon has not told the public the truth about what she knew and when.
“We cannot set a precedent that a First Minister of Scotland can mislead the Scottish Parliament and get away with it.
“We have to trust that the First Minister will be truthful. We no longer can.”
