Dr Reilly had pushed for changes to be made to the Budget announcement at a meeting of the cabinet sub-committee on health earlier this week.
His attempts came after it emerged that the tax relief cut drifted under the radar as it was being looked after by a different government department.
Although his concerns are backed up by some in Labour, the Health Minister was "shot down" when he raised the issue with Mr Noonan at the meeting.
And at the launch of the Government's strategy to tackle alcohol misuse, Dr Reilly said he would not be looking for any further changes to the move.
"No, they are taxation measures passed in the Budget," the Dublin North TD said. "The Dail has passed one so we'll have to implement them."
The change is set to see rises of between €50 and €800 a year in the cost of policies, while insurers are set to impose separate hikes of 10pc in the coming weeks.
Sources in the junior coalition partner say it drifted under the radar because it was being looked after in Mr Noonan's department. One source suggested it was a result of splitting the Department of Public Expenditure and Reform off from the Department of Finance.
Labour minister Brendan Howlin is in charge of the Department of Public Expenditure and Reform.
There has been widespread doubt over whether Dr Reilly can achieve the €666m in savings outlined for his department in the Budget, and a team of senior civil servants is verifying the figures.
Last night the Social Welfare and Pensions Bill passed the second stage in the Dáil by 79 votes to 44, and will be discussed at committee stage this morning.
It gives effect to social protection measures in the Budget which require legislation, including the scrapping of the bereavement grant and cuts to unemployment benefits for new claimants aged under 26.
The final stages of the Bill are expected to conclude at around 4pm today before moving on to the Seanad.