The First Minister dismissed repeated demands to sack the Health Secretary as he came under pressure at Holyrood.
Scots Labour leader Johann Lamont claimed Mr Neil had "deceived" Parliament by telling MSPs he would play no part in moves to reorganise acute mental health services in Lanarkshire.
However, private emails revealed in yesterday's Herald show Mr Neil had by then already told the health board he wanted previously agreed plans to cut beds at Monklands Hospital, in his Airdrie and Shotts constituency, to be reversed.
The instruction came despite health board warnings that keeping beds at Monklands was not in the best interests of patients.
The revelations prompted calls for him to quit. He was also accused of breaching rules that require ministers to avoid possible conflicts of interest.
Yesterday, Ms Lamont renewed the call for him to quit, telling MSPs: "When Alex Neil became Health Secretary in September 2012, he ordered that the changes (to services at Monklands Hospital) be reversed against medical and patient opinion.
"He then deceived this Parliament and deceived the people of Scotland by saying he would take no part in the decision he had already made.
"If Alex Neil won't resign for deceiving Parliament, will the First Minister sack him?"
She challenged Mr Salmond four times - and on each occasion he answered: "No."
The First Minister said Ms Lamont had made herself and her party look "ridiculous" with her repeated calls for Mr Neil's resignation or dismissal.
He said he had considered the case, which dates to 2012, in response to earlier questions about Mr Neil's involvement.
"I looked at the issue carefully last year," he said. "I came to the conclusion Mr Neil had acted perfectly properly.
"Labour's interest in mental health in Lanarkshire isn't anything to do with the patients, it is just an argument to try and get at an SNP minister."
Interviewed on ITV Border later, Mr Neil said the claims against him were "nonsense". He added: "I won't be resigning, absolutely not."
The calls for Mr Neil to quit followed the release of private emails obtained by Labour MSP John Pentland after an 18-month Freedom Of Information battle.
They showed that on September 26, 2012, three weeks after becoming Health Secretary, Mr Neil told NHS Lanarkshire he was "clear in his view" that acute mental health facilities should be retained at Monklands.
His officials added: "The Cabinet Secretary has asked that you seek agreement from NHS Lanarkshire to reconfigure their plans accordingly."
Later that day he told MSPs NHS Lanarkshire was "looking at the future of mental health provision" and added: "I believe it is revising its original proposal for the mental health service at Monklands with a view to retaining an acute mental health facility at the hospital."
But a few hours after addressing Parliament Mr Neil agreed with the head of the NHS that junior minister Michael Matheson should handle the shake-up to avoid "a perception of a conflict of interest".
He subsequently told MSPs the matter was being dealt with by Mr Matheson.
Last year NHS Lanarkshire announced new arrangements for mental health services, retaining beds at Monklands.