Patients First, the organisation set up to make the NHS more open and accountable, issued its call following claims the health board was trying to gag a consultant psychiatrist who had raised concerns about a specialist Mother and Baby Unit (MBU) in Livingston.
The whistleblower, Dr Jane Hamilton, had warned that shortcomings in the service meant people could die. Two mothers subsequently took their own lives on leaving the unit and the family of one of the victims is now suing the health board for medical negligence.
Patients First is also challenging the First Minister's defence of NHS Lothian in which he said all the issues raised had been fully investigated by an independent inquiry.
Dr Hamilton has been suspended or off work with stress-related illness for four years, and told she will not get her job back.
In an open letter to ministers published in The Herald today, Dr Kim Holt, chairwoman of Patients First, and its director, Roger Kline, claim the inquiry commissioned by the health board which Mr Salmond referred to last week, "specifically did not investigate Dr Hamilton's concerns, despite her having lodged formal complaints regarding many aspects of the service".
They say the terms of reference NHS Lothian gave to Dr Margaret Oates, consultant psychiatrist at Nottinghamshire Healthcare NHS Trust commissioned by NHS Lothian, excluded these complaints in particular.
They say it was wrong NHS Lothian was asking Dr Hamilton to sign a compromise agreement that would bring her employment to an end in return for a six-figure sum, but only if she withdrew her clinical concerns.
Patients First has now called on Health Secretary Alex Neil to commission a public inquiry into the cases of current and historic whistleblowers in NHS Lothian.
Dr Oates was sent a copy of the Patients First letter, but did not respond.
Dr David Farquharson, medical director at NHS Lothian said: "The claims being made by Patients First are simply untrue. The investigation carried out by the independent panel including Dr Margaret Oates addressed the concerns raised by Dr Hamilton and found these to be unsubstantiated.
"Dr Hamilton had previously been given the opportunity to raise concerns directly with Dr Oates as part of a peer review of the unit carried out in 2008 and chose not to."
But Dr Hamilton insists concerns were raised and minuted at a very brief meeting in 2008. "If they were investigated, why did have we not seen any outcomes?" she said.
A Scottish Government spokesman said ministers had received reassurances from both NHS Lothian, and the NHS's Central Legal Office that the agreement would not gag Dr Hamilton.
He said an investigation had found the MBU unit functioned well and there was no evidence to support the allegations made. It had also been awarded a rating of "excellence" by the Royal College of Psychiatry.
Meanwhile freelance journalist Suzy Bashford has described it as a tragedy that "a fantastic doctor like Dr Hamilton, with such integrity" had been prevented from working for the last four years.
She said it was the first time she had spoken publicly about time she spent in hospital with suicidal thoughts, but felt she had to having heard of Dr Hamilton's fate.
She said: "Dr Hamilton is the most caring, empathetic, compassionate psychiatrist I have ever encountered."