James Kelly is attempting to repeal the Offensive Behaviour at Football and Threatening Communications Act.
The consultation on his Members' Bill attracted 3,248 responses.
Mr Kelly said 71% were "fully supportive" of repealing the section of the law covering behaviour at football.
And he said there was 62% support for scrapping the section covering offensive communications.
Mr Kelly said this compared with 24% who indicated they were fully opposed to repealing the football part of the act, and 21% the communications sections.
The vast majority of those who responded were individuals, many of whom expressed concerns that the legislation was infringing on freedom of speech and expression and unfairly criminalising fans.
Several people appeared to misunderstand the question, and indicated they were "fully opposed" to repealing the act before later suggesting they were in favour.
Labour said these respondents had subsequently been given an opportunity to clarify their position, with the final figures taking these clarifications into account.
Celtic was the only football club to respond, with its response saying that it "remains supportive" of the Scottish government's commitment to addressing unacceptable behaviour.
But it also said it retained "significant concerns" over potential discrimination against football supporters and confusion in applying and enforcing the law.
Of the 21 organisations that responded, ten were supporters groups including the Celtic Trust, Club 1872 and Fans Against Criminalisation - all of which were in favour of repeal.
Glasgow City Council also indicated it believe the law should be repealed.
But LGBT campaign groups Stonewall Scotland and the Equality Network said repealing the act would send a message that prejudiced and threatening behaviour at football is acceptable, with Stonewall calling for a review of the legislation to examine whether improvements could be made.
The act, which came into force in 2012, criminalised offensive and threatening behaviour, including sectarian behaviour, related to football matches and any communications containing threats or incitement to religious hatred.
The SNP used its majority in the last Scottish Parliament to pass it - but Labour, the Conservatives, Liberal Democrats and Greens all back its repeal.
Mr Kelly said: "The SNP were arrogant to bulldoze this piece of legislation through Holyrood in the first place. Every other party opposed it.
"Academics, lawyers, football clubs and football fans opposed it, yet the SNP wouldn't listen and used their then majority in the Scottish Parliament to rail-road the Football Act through.
"Having lost that majority, and faced with clear public support for repeal through the consultation process, it would be incredibly arrogant if the SNP do not now think again."
But a spokeswoman for the Scottish government said the country continues to have a problem with abusive behaviour at football games.
She said: "A hardcore minority is souring the atmosphere for the majority of football supporters and critics of the Act seem to think our only option is just to accept this contempt for fans and players.
"Not one viable alternative to dealing with the unacceptable scenes of violence and abuse we continue to see at matches has been put forward in the entire debate around this law."
She also said two-thirds of the charges under the law last year were for threatening behaviour, including physical violence.
And she said an independent evaluation had found that the clear majority of fans condemn abusive behaviour towards people's religious beliefs.
The Scottish Conservatives said they would be using part of their parliamentary business on Wednesday to debate the legislation - which could see Holyrood officially vote against its principles for the first time.
The Conservative motion will urge the Scottish Government to "repeal the Act as a matter of priority".
Tory justice spokesman Douglas Ross said: "Sectarianism should not be tolerated under any circumstances, whether it is in our schools, on our streets or in the football stands.
"But the Offensive Behaviour Act is not the way to tackle it.
"It's high time the SNP read the writing on the wall and repealed this deeply unpopular and unnecessary piece of legislation."
The Scottish Greens said they would be working with other parties to "ensure that the act is repealed as quickly as possible".
The party's John Finnie said: "The issues of homophobia and transphobia also need tackled in football, but with a society-wide approach.
"Let's make legislation which treats everyone equal before the law, otherwise we're dealing with forms of discrimination by only targeting people who are football fans."
The Scottish Liberal Democrat justice spokesman Liam McArthur said the football act had been a "knee-jerk reaction" that "puts our civil liberties at risk".
He added: "Ministers must instead move towards a more holistic approach to tackling sectarianism which involves broader action, working with communities, local authorities, churches, football clubs and more."
The majority of responses to a consultation on Scotland's football sectarianism laws are in favour of scrapping the legislation, the Labour MSP behind the move has said.