Mr Martin would not confirm whether impeachment proceedings will be brought before the Houses of the Oireachtas to deal with the controversy.
The Taoiseach is briefing other party leaders this afternoon about the matter.
The Irish Government received legal advice after the Chief Justice said the judge should resign over his attendance at a controversial golf dinner in Galway.
There’s a reluctance for the political system to embroil itself in the judicial arm of the State
The Government reviewed the correspondence between Mr Justice Frank Clarke and Mr Justice Woulfe, which was published on Monday.
Mr Martin said the meeting with opposition leaders is to “get a sense” of how they view the matter.
“I have to be extremely careful about what I say because if a motion of impeachment were to happen, I would be juror in that as an individual member (of the Oireachtas),” he told RTE.
“It is very serious, it’s grave.
“The separation of powers is something I hold very dearly and it’s a crucial part of our democracy.
“There’s a reluctance for the political system to embroil itself in the judicial arm of the State.”
He said the matter will be decided by the Oireachtas.
During a meeting the Chief Justice told the judge he should resign, a position he then repeated in correspondence between them.
The Ceann Comhairle Sean O Fearghail warned TDs this week about making any comments about the controversy, over concerns it could be viewed as bias.
Attorney General Paul Gallagher outlined “serious constitutional issues” arising out of the controversy to Cabinet members.
Meanwhile, Mr Martin said he was not told that a number of other judges had written to the Government expressing an interest to fill the seat on the Supreme Court before Justice Woulfe’s appointment.
It was reported by the Irish Times that senior judges has expressed an interest in the role, however the Cabinet was not made aware.
Minister for Justice Helen McEntee put forward Mr Woulfe’s name to Cabinet in July.
“As far as I am concerned the appointment of judges should not be for political negotiation,” Mr Martin added.
He added that the Judicial Appointments Advisory Board (JAAB) put forward Mr Woulfe’s name.
Labour leader Alan Kelly called on the justice minister to make statements on the appointments process.
He said the decision to appoint Mr Woulfe raises many questions and that if cabinet ministers had not been informed that there were other applicants the Labour party would “consider that to be a very serious issue”.
“It’s quite obvious that the Cabinet – and remember this is a Cabinet decision, this is not a minister for justice decision – that Cabinet members were not informed of the fact that other judges, three other judges had applied for this position. We consider that to be a very serious issue,” he said.
“The fact that Fianna Fail and the Taoiseach are on record, according to the paper, [The Irish Times] that they were not informed either that there were other applicants, we find to be a really serious issue.”
He added it is a “very serious situation” for the Taoiseach, the Justice Minister and the Government.
Article 35.4 of the Constitution of Ireland says that a judge of the Supreme Court or the High Court shall not be removed from office except for “stated misbehaviour” or incapacity, and only upon resolutions passed by the Dail and by the Seanad calling for his removal.
However the Article does not set out the process by which the Oireachtas can pass the resolution.
Mr Justice Woulfe, a former attorney general who was appointed a judge of the Supreme Court in July, faced criticism after it emerged he was among 81 guests who attended the Oireachtas Golf Society dinner in August.
The chief justice met Mr Justice Woulfe last week as part of a resolution process emanating from a report investigating Mr Justice Woulfe’s decision to attend the golf event amid coronavirus guidance against large indoor gatherings.
I think that creates clearly a problem and clearly an issue around tenability of Justice Woulfe's position
Mary Lou McDonald, Sinn Fein
In a letter to Mr Justice Woulfe, Mr Justice Clarke outlined his concerns over how the judge has handled the incident.
Sinn Fein president Mary Lou McDonald said she hopes party leaders will be “fully and properly briefed”.
Ms McDonald repeated the party’s position that Mr Justice Woulfe’s position is “untenable”.
“The publicly known facts are that the Chief Justice has suggested by letter that Justice Woulfe should resign,” Ms McDonald added.
“I think that creates clearly a problem and clearly an issue around tenability of Justice Woulfe’s position.
“I think where the Chief Justice has lost confidence then yes, I would have thought that is manifestly the case.”