Labour made the direct challenge a week after accusing the senior SNP politician of misleading Parliament in 2012 over plans to remove acute mental health beds at Monklands Hospital, which is in his Airdrie and Shotts constituency.
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 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.
"Now that it has emerged the wards he ordered to save contain asbestos, his position has been made even more difficult."
The motion of no confidence will be debated by MSPs tomorrow afternoon. Tory MSP John Lamont said his party would vote with Labour.
Scottish Labour leader Johann Lamont first 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 that 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.
The motion of no confidence in Mr Neil is only the third in the Scottish Parliament's 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.