Andrew Grice: If Daniel Hannan were a lone loose cannon the Tories would be fine. Unfortunately for Cameron, he's not

"We have to mind the gap," one Shadow Cabinet minister told me. He was referring to the danger of the voters perceiving a gap between David Cameron and a Conservative Party which does not always share his modernising zeal.

Leading from the front is necessary in our increasingly presidential system. But the public still votes for a party as well as a leader. It wants to know a bit about the team as well as the captain, and to trust its instincts. This is why internal Tory tensions over the party's attitude to the National Health Service matter. It also explains why Mr Cameron made an unscheduled speech on the issue this week between his holidays in France and Greece.

He may have dismissed as "eccentric" Dan Hannan, the live-wire Eurosceptic Tory MEP who sparked the row by dismissing the NHS as a "60-year mistake" on American television as he entered the passionate debate about President Obama's health reforms. But Mr Hannan is not alone in questioning Mr Cameron's unwavering support for the NHS. Privately, some Tory MPs feel that it is out of kilter with the Tories' vision of a "post-bureaucratic age".

The doubters acknowledge the political risk of the Tories not being committed to the NHS, which was rightly described by the former chancellor Nigel Lawson as "the closest thing the English have to a religion". They understand why Mr Cameron describes it as his "number one priority". They believe that his undoubted personal commitment to the NHS, after his life-changing experience with his late son Ivan, will reassure voters.

Yet many Tories, including some members of the Shadow Cabinet, privately question Mr Cameron's decision to promise inflation-plus rises in health spending come what may. Despite Labour's rhetoric, the Treasury has not yet allowed the Government to match it, although it probably will before the general election. For now at least, Mr Cameron's position on health spending is to the left of Gordon Brown's.

It might have been the right policy in good economic times. But the Tory doubters worry that it will confuse and baffle the public as the party fights an "age of austerity" election, promising big spending cuts to reduce the debts it would inherit from Labour. They concede that public fears about the Tories' commitment to the NHS damaged the party at the last election, which is why Mr Cameron swiftly dumped their policy to subsidise NHS patients who opt for private treatment.

But they worry, as one senior Tory put it, that Mr Cameron is "fighting the last war". They suspect that voters have moved on since 2005 – that while they still "love" the NHS, they no longer believe it should be immune to financial pressures or reform. Like Labour, the Tories will bang on about value for money and the old chestnut of efficiency savings. And Mr Cameron, in an important and well-argued speech on Thursday, said he wanted to go "further and faster" than Labour on extending patient choice. Yet ring-fencing the budget sends a signal to the voters that health will be under less pressure than other public services.

Now, it is true that some of the grumbling comes from shadow ministers jealous of Andrew Lansley, the shadow Secretary of State for Health, whose budget enjoys protected-species status while they must sharpen the axe. But it goes wider than that. Some Tories ask whether it might be better to "invest" in the nation's future by giving priority to education rather than preserving an NHS structure that smacks of post-war planning and the top-down government that Tories want to dismantle.

There is also some whingeing that such a landmark policy was handed down by Mr Cameron and George Osborne, the shadow Chancellor, without any debate. It is clearly not up for negotiation and will be the first spending pledge in the Tory manifesto. The only other budget guaranteed to rise is international development, and that could be fudged once the Tories were in power by including money spent on climate change and defence.

Some Tories suspect Mr Cameron will get little political payback for his "investment" in health – that a public increasingly sceptical of all politicians' promises will not judge his pledge to boost spending as credible. His allies are unrepentant, saying Labour would like nothing more than for him to vacate the battleground.

Although the Tory leader insists he is not looking for a "Clause 4" moment, he would love his support for the NHS to symbolise how the Tories have changed, in the same way that Labour's decision to dump old-style state ownership convinced voters the party had changed its policies as well as its leader. He may be disappointed.

Team Cameron insists their man relishes the opportunity created by the Hannan row. I am not so sure. The controversy was badly timed, jarring with the Tories' drive to portray themselves as the "true progressives" of British politics. Their initial reaction was less convincing than Mr Cameron's more considered speech. Much as they would love to cruise to election victory without much scrutiny, this affair is a warning that, sooner or later, the media spotlight will turn on their policies – possibly with a vengeance.

Although normal political weather may resume shortly, it has been a rather sticky summer for the Tories. The health controversy has given a demoralised Labour Party a much-needed glimmer of hope. To have any chance of getting back in the game, Mr Brown needs Tory policies in the headlines and, in effect, the campaign to start now. "We can't leave it until the election, as Cameron wants," one minister said. "That would be too late. By then it would be a done deal."