Labour’s Henry McLeish ‘s comments come after Mr Salmond said there was “no doubt” Ms Sturgeon broke the ministerial code.
Giving evidence at the Scottish Parliament committee inquiry into the Scottish government’s botched investigation into allegations of sexual harassment against him last week, Mr Salmond stopped short of calling for Ms Sturgeon’s resignation.
Appearing before the committee on Wednesday, Ms Sturgeon rejected his accusations and said she felt “let down” by his “absurd” claims of a plot of SNP figures against him.
Speaking to BBC Radio Scotland, Mr McLeish said: “The First Minister, I think, has rebutted most of the challenges, the assertions, the allegations that have been made.
“In my view there is no serious path to the First Minister either resigning or suffering with a vote of no confidence in the parliament.
“What we should be doing now is for both the committee of inquiry at Holyrood and the separate inquiry into the breach of the ministerial code to be completed as soon as possible, get on with the election and get Scotland back to some normality.”
He said “two distinguished, prestigious people” can’t continue to be seen “knocking hell out each other in public”.
“That’s got to be left behind,” he said.
Mr McLeish continued: “I hope at the end of all of this the parliament and the government learn lessons. That’s the important thing. There are reforms required and that should be the first priority after we get this initial mess sorted out.”
Ms Sturgeon told the cross-party committee of MSPs that Mr Salmond had been one of the “closest people to me in my entire life,” and that she “would never have wanted to ‘get’ Alex Salmond.”
Ms Sturgeon said: “I would never, ever have wanted any of this to happen. I had no motive, intention or desire to ‘get’ Alex Salmond.”
Mr Salmond was acquitted of 13 charges of sexual assault following a criminal trial last year.
The committee was set up after a successful judicial review by Mr Salmond resulted in the Scottish Government’s investigation being ruled unlawful and “tainted by apparent bias.”
A separate inquiry is investigating if Ms Sturgeon breached the ministerial code, which she denies.
