The MEP suggested that the payments would help fund the cash-strapped National Health Service.
But his proposal was immediately ruled out by Andrew Lansley, the shadow health secretary, who said that the system would cost more to run than it would provide in revenue.
It is the second time in a month that the Tory leadership has had to distance itself from comments on the NHS made by Conservative MEPs.
Mr Cameron faced calls to sack Daniel Hannan after the MEP went on television in
After Labour seized on the row to question the Conservatives’ commitment to free health care, Mr Cameron described Mr Hannan’s views as “eccentric” and made clear his own strong support for the NHS.
Dr Tannock, who worked as a consultant psychiatrist before being elected as an MEP, outlined his suggestion during a discussion on Channel 4 News.
He said: "I would be totally in favour of small co-payments, small payments being made if you turn up to things and perhaps small fines being levied if you do not.
"I know they are controversial but I don't think people who are in a job would be against say spending £10 to see their GP or being fined £10 if they don't show up to an out-patients, so that's the sort of thing I would like to see."
Dr Tannock also suggested that doctors should face a pay freeze.
He said: "Right now the doctors in the NHS are the most highly paid doctors in the public health services in the whole of
But Mr Lansley said that the "great majority" of NHS health care should be "based on our need not our ability to pay".
He added: "Charles may say 'oh well £10 to see a GP' - well that would not make any difference to hospital care and frankly, if you tried to put in such a system, by the time you have done all the bureaucracy of raising the money and levying some fines, you would not be raising any money for the NHS anyway."
