Ms Gallagher was booted from the Senate on Wednesday, after the High Court ruled she was ineligible to sit in the upper house as she was still a British citizen when she nominated for election in 2016.

However, Ms Gallagher entered the Senate in March 2015, after being appointed by the ACT Legislative Assembly to fill the casual vacancy created by the departure of Kate Lundy.

On Thursday, Labor and the Greens voted down a Liberal motion to set up a privileges committee to look at whether the Assembly had been misled when it backed Ms Gallagher's nomination for the Senate in 2015, and her statutory declaration that she was not a foreign citizen.

An Assembly committee already looked at the Assembly's two past nominations for Senate casual vacancies, including Ms Gallagher's nomination, finding the ACT's parliament had the most "robust" procedures to select a senator in the nation, Chief Minister Andrew Barr said.

He said section 44 of the constitution covered a wide range of matters, including whether someone was an undischarged bankrupt or insolvent, or had a pecuniary interest in the public service, and it was the role of the High Court - not the Legislative Assembly - to determine whether those factors made someone ineligible for elected office.

"We are simply not in a legal or practical decision to do so," Mr Barr said.

But Canberra Liberals leader Alistair Coe said there needed to be changes to ensure the Assembly did not endorse an ineligible candidate again.

He said he would "much rather we discard the statutory declaration process we currently have rather than do it and do it poorly".

"As it currently stands, the only responsibility for our legislature is to ensure [the new senator] comes from the same party as the departing senator," Mr Coe said.

"If that's the only responsibility we're going to take on, well let's just do that. Let's not take on any additional responsibility. However if the Assembly chooses to put in statutory declarations, if the Assembly chooses to provide more information to the Governor-General, let's make sure it's right .

"We're much better off doing a limited number of things well than doing a plethora of things poorly because when you do that, you erode trust when you can't actually authenticate the integrity of what we're saying."

Former speaker and Canberra Liberal Vicki Dunne said the ACT parliament inadvertently misled the Governor-General by nominating Ms Gallagher, and an inquiry into what went wrong would go some way to amending that.

"Perhaps we write a letter [to the Governor-General] saying, ‘we stuffed up, we’ve looked at our procedures and this won’t happen again’," she said.

"We did make a mistake. We did appoint someone who it turned out, with hindsight, [was ineligible] and hindsight makes us very wise. Now we have that wisdom let us look at our processes and make sure we get it right in future."

ACT Labor secretary Matt Byrne has defended the party’s role in Ms Gallagher’s nomination, and also committed to review the party’s processes.

"I imagine that the party at both the territory and federal level will have to review our processes because of the ruling," he said.

"The party took every step it could to ensure that all our candidates are eligible including when the vacancy in the Senate occured. The party has been improving, well we thought we'd been, the processes, for a number of years now."

"I accept that clearly there was some things between the federal party's processes and the territory's one that we didn't get one hundred per cent but again if you go to Katy's statement, she acted on the advice she was given and in the 2016 vetting process it was everyone was acting on the view at the time that she had done what was required."

Before the High Court ruling, Ms Gallagher had maintained she was not a dual citizen.

She first came under the citizenship cloud last August, when it was first suggested that her mother may have been an Ecuadorean citizen.

However it was because of her father's British citizenship that Ms Gallagher came unstuck.

While she renounced any entitlement to British citizenship just before the 2016 election, on advice from Labor officials, the senator had already been in parliament for more than a year.

Ultimately the High Court ruled she was still a British citizen when she nominated for the Senate in 2016, thus rendering her ineligible to sit in the Australian Parliament and sparking a raft of resignations from five other MPs under a similar cloud.