It comes after Nicola Sturgeon confirmed her predecessor faces two complaints of harassment as he pursues legal action against the Scottish government.
The Scottish first minister admitted her relationship with Mr Salmond has left her in an “extremely difficult situation” and the case would be “upsetting” for the SNP.
But, she added the complaints “could not be ignored or swept under the carpet”.
Mr Salmond is seeking judicial review of the investigation into the complaints, which he denies.
He has characterised parts of the claims as “patently ridiculous”.
The ex-first minister and former MP has attacked the “unlawful procedure” against him, lamenting that the investigatory process being used had been introduced after he left office.
He said he wanted a “real court” to decide on the appropriateness of the process.
Ms Sturgeon replaced Mr Salmond as first minister and SNP leader following the 2014 Scottish independence referendum, with the his successor acknowledging the “length and the closeness” of their relationship.
In a statement on Friday, she confirmed complaints were made against Mr Salmond by two individuals in January, at a time of a wider focus on harassment within UK politics and other industries.
It is reported they date to his time as first minister, with one alleged incident to have taken place in the first minister’s official Edinburgh residence in December 2013.
Ms Sturgeon said: “These complaints have been considered since then under a procedure covering ministers and former ministers that was agreed by me in December 2017 in the wake of public concern about harassment.
“Although I have been aware for some time of the fact of the investigation – initially from Alex Salmond – I have had no role in the process, and to have referred to it before now would have compromised the integrity of the internal investigation, which I was not prepared to do.”
Ms Sturgeon said the Scottish government “refutes” Mr Salmond’s criticisms of its process and “will defend its position vigorously”.
“However, this focus on process cannot deflect from the fact that complaints were made that could not be ignored or swept under the carpet,” she added.
“I have been clear on many occasions that all organisations and workplaces must make it possible for people to come forward to report concerns and have confidence that they will be treated seriously.
“For that principle to mean anything it cannot be applied selectively. It must be applied without fear or favour, regardless of the identity, seniority or political allegiance of the person involved.
“My relationship with Alex Salmond obviously makes this an extremely difficult situation for me to come to terms with. I am also acutely aware how upsetting this will be for my party.”
In a statement late on Thursday, Mr Salmond revealed his legal action.
He described how he had attempted to persuade Leslie Evans, permanent secretary to the Scottish government, that “she is behaving unlawfully in the application of a complaints procedure introduced by her more than three years after I left office”.
“This is a procedure so unjust that even now I have not been allowed to see and therefore to properly challenge the case against me. I have not been allowed to see the evidence,” he added.
Revealing he will “say nothing which would impinge on the court proceedings”, Mr Salmond continued: “If I lose then I will have to answer to the complaints both comprehensively and publicly.”
But, if he wins his case, Mr Salmond claimed “the administration at the senior levels of the Scottish government will have the most serious questions to answer”.
Ms Evans said Mr Salmond’s statement “contains significant inaccuracies”, adding: “As the head of the civil service in Scotland, I have been consistently clear that there is no place for harassment of any kind in the workplace.
“In line with work already under way in the organisation to tackle inappropriate behaviour, I will consider carefully any issues about culture and working practices highlighted by these complaints.”
Police Scotland had no comment on the claims, a spokesman said.
After losing his Westminster seat at last year’s snap general election, Mr Salmond controversially went on to host a show on Kremlin-backed TV channel Russia Today.