The First Minister conceded that adviser Campbell Gunn made a "mistake and a misjudgement" when he contacted a newspaper with claims about Clare Lally.
She is the mother of a disabled girl, and recently spoke at a Better Together event.
Mr Salmond came under pressure from Labour, the Conservatives and the Liberal Democrats to sack Mr Gunn - with those calls reiterated after the First Minister's spokesman admitted Mr Gunn's email was "clearly not appropriate".
Now Labour's Holyrood business manager Paul Martin has written to Sir Peter Housden, Permanent Secretary to the Scottish Government, urging him to intervene.
He said: "The First Minister failed to take action against one of his most senior advisers for trying to undermine a member of the public because of her views on the referendum."
Ms Lally, 33, was the target of online abuse after she told a Better Together campaign rally she was "an ordinary mum from Clydebank campaigning for Scotland to stay in the UK".
Mr Gunn apologised "unreservedly" to her after it emerged he had emailed a newspaper to advise that Ms Lally was a member of Labour's shadow cabinet and then incorrectly said she was related to former Glasgow Labour Lord Provost Pat Lally.
Opposition leaders insisted that meant Mr Gunn had breached the code of conduct for special advisers, which states that advisers must not disseminate personal attacks and inappropriate material.
Mr Martin, who has also written to Sir Bob Kerslake, head of the UK civil service on the issue, said Mr Gunn's email was meant to "undermine the credibility of a member of the public, Clare Lally" and added: "It was clearly a personal attack in contravention of the special advisers' code."
Mr Martin claimed Mr Gunn had relied on a website which is "deeply offensive in nature" and which "peddles untruths and rumours as the truth" when he stated incorrectly that Ms Lally was the former lord provost's daughter-in-law.
In his letter to Sir Peter, the Labour business manager said: "Mr Gunn's actions were clearly party political - a breach of the code."
A Scottish Government spokeswoman said: "We will respond in due course to the letter from Paul Martin."