Mr Neil said that people deciding how to vote in next month's independence referendum were "very frightened of the implications of a No vote for the future of the National Health Service in Scotland".
Pro-independence campaigners have repeatedly claimed a Yes vote is needed to protect the NHS in Scotland - but this is fiercely denied by those who want Scotland to stay in the UK, who point out that decisions over health are already devolved to Holyrood.
Former first minister Jack McConnell has branded nationalist claims over the NHS a "big huge lie", adding: "'It is a diversion, a desperate tactic. Having criticised others for negative campaigning, it shames the Yes campaign."
But Mr Neil said: "We're telling the facts, we're sticking to the fact. The facts are the future budget for the health service is under serious threat if we decide to stay in the UK.
"The only way we can guarantee the health budget and the founding principles of the health service will be protected in Scotland will be a Yes vote."
The Health Secretary spoke out as he visited an opticians in the Toryglen area of Glasgow with Dr Willie Wilson, one of the founder members of the pro-independence group NHS for Yes.
Mr Neil said people in Scotland "see the scale of privatisation south of the border, they see the charges that are being introduced". He added: "People are frightened by that prospect."
He stated: "By far the biggest concern is over the health service budget. As a devolved government our health budget is effectively decided as a percentage of the health service budget in England."
But he said the NHS in England was being privatised by Westminster so that "in the medium to long term they can reduce public funding on health".
That would have a knock-on effect on health spending in Scotland, Mr Neil claimed, saying: "If public funding is reduced the impact on the budget in Scotland under the devolved system could be very detrimental indeed.
"In an independent Scotland we would be entirely in charge of the decision of how much we allocate to the National Health Service in Scotland.
"At the moment we get a budget from Westminster that is fixed, we've no taxation powers and no borrowing powers.
"The only way you can protect the health budget would be by reducing spending on education, on housing, on all the other things."
The Health Secretary went on to claim both the Conservatives and Labour were threatening to cut billions of pounds from public spending in the UK after the 2015 election.
Mr Neil said: "You cannot put additionally, on top of the existing cuts, another £25 billion of public spending cuts and still expect the health service to be fully protected.
"If you protect the health service after £25 billion of cuts, it means the cuts in education and other essential services would be even more dramatic."
Dr Wilson said: "By voting Yes in the forthcoming referendum we can secure the progress that has been made in health under devolution and take the opportunity to protect NHS Scotland from Westminster's destructive privatisation and austerity agenda."
He said the NHS in England had undergone an "avalanche of privatisation", adding: "In Scotland, we have not gone down this road - but the more Westminster cuts Scotland's budget, and the more the state withdraws from direct NHS provision in England, the greater Scotland's budget will be squeezed as a consequence because of the way the Scottish Parliament is currently funded."
Optometrist Tahra Mahmood said her work is exclusively funded by the NHS, adding: "A public health service, and in particular free NHS eye examinations, are incredibly important to my patients in Toryglen.
"A Yes vote allows us to safeguard the NHS against cuts that come from Westminster.
"This referendum is a choice between two futures - one of austerity where we waste money on nuclear weapons and one where decisions about Scotland's future are in Scotland's hands."