Ms Burton said she had confidence in the Fine Gael health minister but only after being pushed several times on the issue by Fianna Fáil.
After Sinn Féin then claimed she was “standing by her man”, she responded: “I am no Tammy Wynette”.
The exchanges occurred in the Dáil yesterday after Ms Burton gave formal notice to the House of junior health minister Roisín Shortall’s resignation.
Ms Burton was standing in for Tánaiste and Labour leader Eamon Gilmore, who is attending the UN General Assembly in New York.
Fianna Fáil TD Michael McGrath said it was clear that Mr Gilmore had “shafted” Ms Shortall to protect Dr Reilly.
He also said that while some in Government might seek to claim Ms Shortall’s resignation resulted from a personality clash with Dr Reilly, “nothing could be further from the truth”.
Instead, Mr McGrath said, she had resigned after making a “devastating critique” of the record of Dr Reilly and the Government on health.
“[She] was forced to articulate in public her frustration at the failure to prioritise the extension of free GP care and the lack of transparency in the selection criteria used for locating new primary care centres,” said Mr McGrath.
“The tipping point came when Minister Reilly discarded the established criteria for locating new primary care centres and failed to explain why two centres in his constituency jumped the queue ahead of everybody else.”
He questioned if Ms Burton and the Cabinet had been consulted on Dr Reilly’s decision to add 15 locations to the original list of 20 primary care centres.
He also questioned if Ms Burton had full confidence in Dr Reilly.
Ms Burton eventually said she had confidence in Dr Reilly just as she had confidence in all members of the Cabinet.
Sinn Féin TD Mary Lou McDonald criticised Ms Burton for “standing by her man” when “anybody with an ounce of wit” could see Dr Reilly was “not a suitable occupant” of health.
But Ms Burton defended Dr Reilly while also praising Ms Shortall.
Their “joint work” had ensured a “pipeline of projects” that would “transform the face of primary care”.
On the issue of consultation, Ms Burton said Dr Reilly had brought the expanded list to Cabinet but said ministers would not have delved into the “minutiae” of it.
“That is not the way it works,” she said.