Lib Dem MSP Jim Hume said the majority of the 161 responses to his consultation showed an appetite for the crackdown.
The position puts him at odds with Deputy Prime Minister Nick Clegg, the Lib Dem leader at Wesminster, who said enforcing a ban would not work.
Mr Hume wants other MSPs to get behind his plan and help get it through the Scottish Parliament.
"It is clear that many agree with me that these are fair and specific proposals whose time has come," he said.
"Many of the charities, individuals and health professionals who have backed my proposals recognise that this is about giving children the best start in life. It doesn't seem fair that a child should be cooped up in a smoke-filled car during the school run.
"Those children cannot change their means of transport, let alone take steps to immediately remove themselves from the uncomfortable confines of a smoke-filled car."
Mr Hume lodged draft proposals for a Bill in May last year calling for a ban in Scotland on smoking in private vehicles while a child under 16 is present.
Motorists could be fined £60 for breaching the rule.
The consultation responses were split among 88 sent to Mr Hume and 73 completed through an online survey by the British Heart Foundation.
It showed 75% supported the plan, rising to 84% when the survey was included.
Some concern was raised about how officers will be able to identify the age of passengers and the age of the offender. Police would have to look out for cigarettes and children, alongside seatbelts and mobile phones.
One respondent claimed the approach is profoundly illiberal.
The Tobacco Manufacturers' Association and smokers' lobby group Forest said the plan is a disproportionate response to a relatively rare problem.
Mr Clegg's intervention comes the day after it appeared a possible ban had moved a step closer at Westminster with support from peers.
Labour wants to establish UK Government powers to make smoking with children in the car an offence. The move could become law because of suggestions that MPs will be given a free vote on the issue when the Children and Families Bill returns to the Commons.
But Mr Clegg, speaking on LBC 97.3 radio's Call Clegg show, said: ''I don't personally think that it is going to work to pass a law.
''Of course it is a stupid thing to do to smoke in a car with kids in the back, of course it is - in the same way you shouldn't give your children a can of Coke before going to bed or only feed them on crisps breakfast, lunch and supper. I'm like anybody else, I've got small children, I'm dismayed that anyone might do that.''
But he added: ''The question is, 'Is it right always to have a law to fix things you don't like?'. I know the temptation is to say, 'There's a problem, where's the law?', but I am quite an old-fashioned liberal and I think you shouldn't legislate unless you think it is going to make a difference, and I don't see how this is going to be enforced."
Mr Hume received enough cross-party support to formally lodge his proposed Bill within hours of publishing the consultation responses.
"This unprecedented move gives me confidence in the path ahead as the Bill makes its way through Parliament," he said.
"I look forward to working with all members to delivering these fair proposals, which will enable more children to have the healthiest start in life."