Minister for Foreign Affairs Simon Coveney has said he accepts "mistakes were made" in the appointment of Katherine Zappone as Ireland's UN Special Envoy on Freedom of Opinion and Expression.
Earlier, the former minister for children said she would not accept the role.
There had been calls for clarity over the appointment process, with Opposition TDs describing the appointment as "cronyism" and lacking any real transparency.
Ms Zappone's decision to decline the role also follows a report in the Irish Independent that she organised an outdoor gathering of 50 friends and former colleagues, including Tánaiste Leo Varadkar, at the Merrion Hotel six days prior to her appointment as UN special envoy.
Mr Varadkar confirmed that he attended the event and said he accepted "that mistakes were made in the manner in which Dr Zappone was appointed" and regrets "the controversy it's caused".
In a statement, Ms Zappone said: "While I am honoured to have been appointed by the Government to be the Special Envoy on Freedom of Opinion and Expression, it is clear that criticism of the appointment process has impacted the legitimacy of the role itself.
"It is my conviction that a special envoy role can only be of real value to Ireland and to the global community if the appointment is acceptable to all parties.
"For this reason, I have decided not to accept this appointment, and I have communicated my decision to the Minister for Foreign Affairs," the statement added.
Minister Simon Coveney responded: "I note the statement from Dr Katherine Zappone stating she will not now take up a special envoy position.
"I understand her decision and thank her for being willing to work on Ireland's human rights priorities.
"For my part, I accept that mistakes were made in how this appointment was brought to Government and lessons will be learnt from that."
Sinn Féin leader Mary Lou McDonald said: "Now we need accountability from Government for the lack of process/ procedure in this attempted appointment.
"The guest list for the Merrion Hotel function must also be made public. The days of funding jobs for the boys and girls on the inside track with taxpayers' money must come to an end."
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Fianna Fáil Senator Malcolm Byrne said he believes that Ms Zappone "made the correct decision" by choosing not to take up the role as UN Special Envoy.
Speaking on RTÉ's Drivetime, he said that he and "a lot" of his colleagues in the party were "angry and annoyed" at the events surrounding her appointment as "not a good look".
Mr Byrne said: "At a time when the vaccination programme is going well, we're looking at big investments in housing and infrastructure and education, and then suddenly, this story emerges on the scene.
"So from within the party, there's a lot of annoyance."
He said "more answers" are needed on details surrounding the event held at Dublin’s Merrion hotel.
Sinn Féin's Matt Carthy has written to the chair of the Public Accounts Committee asking it to examine the appointment of Ms Zappone to the UN role.
Mr Carthy, a member of the PAC, also called on Ms Zappone to turn down the special envoy role, saying he was not convinced of the "need for the role" and it was a "made up job".
Regarding the event Ms Zappone organised at the Merrion, he said he believed that the spirit of the regulations were "absolutely breached".
When asked about the attendance of the Tánaiste at the event, he said that Mr Varadkar needs to explain the context around his attendance.
He added that the Taoiseach will ultimately decide if there are sanctions.
Mr Carthy said there are "striking similarities in terms of the rationale and excuses that are now being put forward" for this event and the Oireachtas Golf Society dinner in Galway last year.
Earlier Ms Zappone told RTÉ News that she was "assured" by a Dublin hotel that an event she organised at the venue was "in compliance with Government Covid-19 restrictions and guidelines".
In her brief earlier statement, Ms Zappone said that further queries regarding the event should be directed to the hotel itself.
In a statement, the Merrion Hotel said: "At all times since the start of the pandemic, we have adhered to government public health measures, including the relevant guidelines that were in place at the time of the event you reference on July 21st. These have since been updated on July 23rd.
"Government regulations at the time allowed for outdoor events of up to 200 people, and the definition of that includes social events. This upper limit still applies, except now indoor events in hotels are excluded per government guidelines published by Fáilte Ireland."
Cabinet decision caused some 'disquiet'
Separately, Fianna Fáil TD and member of the Foreign Affairs Committee Barry Cowen called on the Minister for Foreign Affairs to appear before the committee to discuss the appointment of Ms Zappone to UN special envoy.
Speaking on RTÉ's Prime Time, Deputy Cowen said the minister should come before the committee, to "highlight the reasoning behind the appointment, the methodology around the appointment, and the decision making process he entered into prior to the nomination being put to the Cabinet last week".
He said he believes the decision was a "bad one".
"There hasn't been a process entered into whereby somebody else might have been invited to come forward to make themselves available, and to give whatever skill set, could have been found by somebody else to match that, if not surpass that of Katherine Zappone."
Deputy Cowen also suggested that the UN ambassador and her staff, who he said are also based in New York, could have been considered.
He said the "manner" in which the appointment was brought to the Cabinet, "surprised" him, in relation to what he said was a lack of information, and a lack of notice surrounding the appointment.
Deputy Cowen said the Cabinet’s decision has caused some "disquiet" among the wider public and the wider political sphere.
Minister Coveney’s account of what happened "wasn’t as inspiring as it could have been," Mr Cowen added.
He said he believes more information might be imminent in a committee setting.
"To put these questions to his own staff within the department. When did they feel that such a role was necessary? What approach was made by whom in relation to that role?"
When asked about allegations of cronyism in relation to apparent events leading to Ms Zappone’s appointment Deputy Cowen said, "that's a problem".
"Perception matters and the perception is that Katherine Zappone made herself available for a role, and that role was provided and offered to her exclusively.
"The public expect in this day and age, that there should be adequate competition, that there are going to be adequate processes and procedures in place that are transparent, that people can stand over, providing the best possible person for the job, irrespective of what remuneration is."
Deputy Cowen said if he were in Katherine Zappone’s position now, he would not accept the role.
In relation to the alleged leaking of this information from Cabinet, Deputy Cowen said the Taoiseach and Tánaiste should meet on this matter and "put in place corrective measures" to ensure that this does not happen.
He said the leaking of information from Cabinet has been "happening all too frequently in recent years".