Ms Gallagher asked for the referral because she was a British citizen on the nomination date for the 2016 election.
In his statement, Mr Falinksi had only said he was not a dual citizen and had sought legal advice to confirm that.
Labor has been dealt a further blow in the citizenship crisis, with Victorian MP David Feeney flagging his status is likely to go to the High Court because evidence can not be found that he renounced his British citizenship.
Liberal MPs Ross Vasta and Nola Marino say they have been told they are not dual Italian citizens, but are also unable to provide paperwork proving their claims.
Labor has secured crossbench support to refer nine MPs to the High Court over their citizenship status.
Coalition MP Ann Sudmalis provided legal advice that as she had been born to a British mother, she was not eligible for automatic British citizenship - unlike those born to British father such as former Bennelong MP John Alexander.
Mr Burke said Labor would make that referral if the documents can not be found.
A vote to refer eight other MPs to the court was narrowly defeated in the House of Representatives.
Labor's Josh Wilson, the member for Fremantle, and Justine Keay and Nick Xenophon Team MP Rebekha Sharkie also fall into the group of MPs who took steps to renounce their British citizenship before nominating but did not have their renunciation confirmed until after nominations closed on June 9.
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He revealed Labor was even prepared to refer to court Environment and Energy Minister Josh Frydenberg, whose parents arrived in Australia stateless after escaping the Nazis.
This could mean they not in the clear, as Mr Turnbull gave his MPs weeks to prepare the needed documents, The Sydney Morning Herald reported.
Attorney-General George Brandis has demanded Labor leader Bill Shorten "do the right thing and refer them himself".
Mr Burke argued they had taken every possible step they could have to renounce their citizenship.
"The government does expect that Mr Shorten will take the step that has been taken by the Leader of the Opposition in the Senate, and refer those four members to the Court of Disputed Returns because if he fails to do so, that raises very serious questions about Bill Shorten's own credibility", Senator Brandis told the upper house after Senator Gallagher's statement.
Ms Wong said Labor did not resile from their belief Senator Gallagher is entitled to be in Parliament.
"The Australian people are sick of this constitutional crisis, the only way to resolve this citizenship crisis is with the cooperation of all sides of parliament", he said.
Senator Gallagher stood aside from her frontbench portfolios and as manager of opposition business in the Senate.