Fianna Fáil says it is 'reviewing all options' in relation to the appointment
Former Attorney-General Máire Whelan is now a judge of the Court of Appeal.
Judge Whelan was formally appointed in a short ceremony at Aras an Uachtarán.
Her warrant of appointment was signed by President Michael D Higgins and countersigned by An Taoiseach Leo Varadkar.
Two judges of the High Court - Eileen Creedon and Charles Meenan - were also appointed at the ceremony.
WATCH: Máire Whelan becomes a Judge, appointed to Court of Criminal Appeal pic.twitter.com/r3ko2GREnw— Páraic Gallagher (@paraicgallagher) June 19, 2017
Ms Whelan's appointment has faced criticism, with Transport Minister Shane Ross saying the nomination should be reviewed 'to reassure people' of its validity.
Fianna Fáil has said the appointment would breach its Government confidence and supply deal.
Ms Whelan was appointed to the role during Enda Kenny's final Cabinet meeting as Taoiseach. The role carries a salary of €180,000 a year.
Her name was put forward by the former justice minister, Frances Fitzgerald, after the Judicial Appointments Advisory Board was unable to suggest any candidates.
Taoiseach Leo Varadkar has said he stands by the appointment, describing Ms Whelan as "the stand-out person for the vacancy".
Speaking earlier on Newstalk Breakfast, Fianna Fáil spokesperson for public expenditure and reform Dara Calleary said the issues around this appointment go further back.
"As the Attorney-General, Máire Whelan was a member of the Judicial Appointments Board that oversaw this process, she as Attorney-General is required under law to declare that she had an interest in any position.
"So if this position was discussed by the Judicial Appointments Board... and she didn't declare that she had an interest - and we understand there was a meeting last May, so presumably she knew then that she had an interest in it - then there is a problem there, too.
"Secondly, there is the issue of the three High Court judges.
"Of the current existing Appeals Court, the majority of them are actually judges that came from the High Court.
"So it is an avenue for High Court seeking other promotions.
"We are not happy the way Frances Fitzgerald dismissed the genuine concerns around how they were treated yesterday, and we'll be seeking her to come in to the House and discuss what exactly happened.
"The three High Court judges, in accordance with the current procedures, had to express their interest to Márie Whelan - who eventually got the job".
Asked if this could collapse the confidence and supply deal, Mr Calleary said: "We're going to review all of our options in relation to this".
Additional reporting: Páraic Gallagher