Opposition politicians have been calling for the inquiry following the publication yesterday of the latest report into health service failures in the region.
Nine people with lung cancer were misdiagnosed by a locum radiologist. Eight have since died.
The HSE first became aware of the problem in September of last year but it was eight months later before the review of nearly 6,000 X-rays and CT scans actually got underway.
Today, the Department of Health said there would be no inquiry.
Fine Gael TD Fergus O'Dowd has criticised the decision.
He said a letter from a radiologist back in 2001 warned that mistakes would be made and showed that clinical governance was not happening in the hospitals at the centre of this latest crisis - Our Lady of Lourdes in Drogheda and Our Lady's in Navan.
Deputy O'Dowd said maladministration had claimed lives and the public deserved to know why.
