Labour made the direct challenge a week after accusing the senior SNP politician of misleading parliament over plans to remove acute mental health beds at Monklands Hospital in Airdrie and Shotts in 2012.
The party revealed emails they say shows Mr Neil had intervened in the decision-making process after becoming Health Secretary, before stepping back and handing responsibility to his deputy, the public health minister, citing concerns over a "conflict of interest".
Labour MSP Neil Findlay, who lodged the motion, said: "Despite all the evidence to the contrary, Alex Neil and Alex Salmond continue to blindly claim that no wrongdoing has taken place with the health secretary's involvement in mental health services at NHS Lanarkshire.
"Nearly a week after these damning emails were released under Freedom of Information, Alex Neil's explanation as to why he meddled in local services and then claimed he had excused himself from the decision have not been satisfactory.
"Medical experts stated that keeping Monklands wards open would result in a 'less than optimal service' for patients.
"Yet Alex Neil has been unable to provide any clear rationale or evidence for changing the decision, which is contrary to his own health policy of delivering health services in communities wherever possible.
"Now that it has emerged the wards he ordered to save contain asbestos, his position has been made even more difficult.
"His failure to come up with answers to the questions being asked of him mean he is not fit to be running our health service."
The motion of no confidence will be debated by MSPs on Wednesday afternoon.
Scottish Labour leader Johann Lamont called for Mr Neil to be sacked during ill-tempered exchanges at First Minister's Questions on Thursday last week.
She ran through a timeline of events which appeared to show his office sent an email stating his view is to retain mental health services in Wishaw and Monklands.
The order reversed previous health secretary Nicola Sturgeon's policy, made on the advice of medical professionals, Ms Lamont argued.
''He then told this chamber, the head of the NHS and the civil service that he would absent himself from a decision on these services, even although Alex Neil had already made a decision," she complained.
Mr Salmond said the Labour leader had made herself and her party look ''ridiculous'' by calling four times for Mr Neil's resignation or dismissal.
''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,'' he said.
The motion of no confidence in Mr Neil is only the third in the Scottish Parliament's short history.
The SNP tabled one in December 2000 against Labour MSP Sam Galbraith, who was education minister during a school exams fiasco.
A second, tabled by the SNP in February 2001, questioned the role of Labour transport minister Sarah Boyack in awarding trunk roads contracts to private firms.
Both attempts failed when put to the vote.
Tory MSP John Lamont said his party will vote with Labour.
"There is a simple principle here - it is not acceptable for a minister to come to the chamber and say things that he knows to be untrue," he said.
"In another time, any minister who misled parliament would be expected to fall on his sword.
"To see the First Minister defend Alex Neil when he's been caught bang to rights, having misled the chamber, is an unedifying spectacle indeed."
Lib Dems will also back Labour, saying the group "takes the matter of lying to parliament extremely seriously".