Catherine Martin's position as Media Minister is "now untenable", the Labor Party said.
Politicians have reacted to RTE board chairman Siun Ni Raghallaigh resigning overnight after Ms Martin failed to express confidence in her during a TV interview.
Ms Martin said she had been "misinformed" about the approval of an exit package for a former RTE executive and was "deeply disappointed" when asked to express her confidence in Ms Ni Raghallaigh.
The Labor Party has accused the minister of "summarily dismissing" Ms Ni Raghallaigh live on air, while the chairman of the public accounts committee accused Ms Martin of "going a bit too far".
Justice Minister Helen McEntee said Ms Martin had not effectively dismissed the RTE chairman during her Prime Time interview.
She said Ms Martin had "outlined the facts as they occurred" and that she "needed and wanted" to speak to Ms Ni Raghallaigh on Friday.
"She didn't fire her. In fact, she has clearly indicated how much work Siun has done in recent months."
RTE has come under heavy scrutiny for releasing details of executive exit packages dating back to 2016.
On Thursday it published legal advice saying that publicly releasing the details of former and current employee contracts would expose RTE "to avoidable legal challenges".
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Questions have been raised over the governance of exit packages after it emerged that former finance director Breda O'Keeffe had been paid €450,000, which was not approved under the terms of a voluntary redundancy programme.
Appearing on Prime Time on Thursday evening, Ms Martin said she had been repeatedly assured that RTE's board had no role in signing an exit package to RTE's former chief financial officer, Richard Collins.
However, it has since emerged that the board's remuneration committee, of which Ms Ni Raghallaigh is a member, approved Mr Collins' severance arrangements.
The minister had said she had requested an urgent meeting with Ms Ni Raghallaigh on the matter on Friday.
However, hours later, Ms Ni Raghallaigh said it was "abundantly clear" her position was no longer tenable because she had lost Ms Martin's trust.
In her statement, issued just before 1 a.m. on Friday, she said her misrepresentation of the trial was not "intentional."
Ms Ni Raghallaigh said she was asked during a meeting with Ms Martin earlier this week whether the board was aware of the exit packages for former strategy director Rory Coveney and Mr Collins.
"I said I was aware of them, but they had not come before the full board. However, I forgot to remember that Richard Collins' exit package was indeed tabled before the compensation committee.'
Ms Ni Raghallaigh said RTE director general Kevin Bakhurst was also present at the two meetings with the minister and her officials this week, which lasted more than three hours.
Labour's Marie Sherlock has said Ms Martin's TV interview amounted to the "summary dismissal" of the RTE board chairman live on air, saying Ms Martin's position was "now untenable" .
She said that from the start of the crisis, Ms Martin has "stayed in the background" and "failed to ask the most fundamental questions".
She said Ms Ni Raghallaigh's resignation statement appeared to reveal details of exit packages of former RTE executives held by the Media Ministry, which she said "should be published in full immediately".
"The Taoiseach and the Tanaiste must now reflect on the way the Media Minister has handled this whole debacle and ask themselves: is she the person to ensure that the necessary reforms take place in RTE that will be crucial to its survival? of it, and is she the person who can restore public confidence in the state broadcaster?"
Brian Stanley, chairman of the Public Accounts Committee, said the minister had "jumped the gun a bit" and "should have kept her powder dry" and waited for the meeting with Ms Ni Raghallaigh.
"The minister, I think, should have waited until she had the opportunity to speak to Siun Ni Raghallaigh and hear exactly what Siun Ni Raghallaigh had to say. And then she could make a judgment about whether she had confidence or not.
He said Ms Ni Raghallaigh had made a "very detailed statement" which raised further questions.
"The minister may have made the right decision, the minister may not have made the right decision, but what I am saying is... there are some lines in (Ms Ni Raghallaigh's statement) that raise questions, and the minister should have waited to sort that out with Siun Ni Raghallaigh to get to the bottom of it.
"If a minister without any notice, I assume, to Siun Ni Raghallaigh or whoever, goes on television and does not express confidence, refuses to express confidence in the head of the board of RTE, then there is a completely new situation."
Media committee member Senator Malcolm Byrne said he does not believe Ms Martin should have conducted the interview before the meeting scheduled for Friday.
"I would have preferred that any decisions would have been taken after consultation between the minister and the chairman.
"If it turns out that the department was aware of the process surrounding Richard Collins' exit package, this will bring the department into this dispute.
He added: "Siun Ni Raghallaigh is a dedicated public servant and has been fair and competent in her dealings with our committee. She and Kevin Bakhurst are trying to reform a very difficult organization."
Ms McEntee said: "Minister Martin has not asked anyone to step aside and we need to be clear about that. She promised to do an interview and didn't follow through. What she did was lay out the facts as they happened this week.
"I think the difficulty here is that we all want to have confidence in RTE, we want to have confidence in those who run it, and we want to move forward. And unfortunately, when information comes out the way it does today on a wide variety of issues, it's very difficult for people to have confidence.
"Of course this will contribute to the challenges that RTE now faces."
Media Minister Catherine Martin thanked Ms Ni Raghallaigh for her service as RTE Board Chair since November 2022.
"At this critical time for RTE, it is imperative that we continue the important process of rebuilding trust in the organisation. As Ms Ni Raghallaigh said, there is an urgent need to continue the transformation process and put in place a sustainable financing model - I am deeply committed to this," Ms Martin said in a statement.
