The White House has denied Republican claims that the rule was withdrawn, saying it was stuck in review on Inauguration Day.
"I don't think it's a long-term path to success," said the Republican congressman, who backed Trump's impeachment in the House.
Thursday's vote sought to remove Greene from her House committee assignments due to her past social media endorsements of several extremist conspiracy theories.
Cheney won support from a majority of House Republicans despite voting to impeach former President Trump.
The House Republican leader voted to condemn the conspiracy theory in October last year.
Cheney saw off a challenge to remove her as the number three Republican in the House with a significant majority.
"I cannot fathom any company would give money to the House Republicans ever again for having a Parkland denier, 9/11 truther in their midst," Wallace said.
The House Minority Leader called Democrats' resolution to remove Greene from her current committee assignments a "distraction" on Wednesday.
"McCarthy's failure to lead his party effectively hands the keys over to Greene — an anti-Semite, QAnon adherent and 9/11 Truther," Speaker of the House Nancy Pelosi (D-Calif.) said Wednesday.
If Democrats fall in line, Republicans will be powerless to stop Greene from being removed from her committee assignments.
The Georgia Republican remains a divisive figure even among Republican voters, with about 1 in 5 ranking her as favorable and unfavorable.
The Republican congressman was criticized for posting on Facebook a cartoon of a teacher lazing in the sun.
Few Republicans have condemned Greene's beliefs which include anti-Semitic, anti-Islamic and baseless claims about school shootings and child abuse.
A new ad from The Lincoln Project accuses House Minority Leader Kevin McCarthy (R-Calif.) or being responsible for the January 6 insurrection at the U.S. Capitol.
The House Rules Committee is seeking to remove the new congresswoman from her positions on the House Education & Labor Committee and the House Budget Committee as early as Wednesday.
House Democrats are pushing McCarthy to draw a harsh line, despite the GOP leader's fears of a splintered party.
"This is not a Trump-first party. This is a country-first party," the Republican congressman said.
Greene supported conspiracy theories about deadly school shootings in comments before running for Congress.
McCarthy is under pressure to remove Greene from her committee assignments following damaging revelations.
The MSNBC host promised viewers "exclusive video" of the meeting but "accidentally" rolled the wrong clip on "The 11th Hour."
The House minority leader discussed the former president assisting Republicans as they look to flip the House and Senate in 2022.
It may not be possible to easily remove Greene from her committee seats if she determines to keep them.
"Once a member is placed on a standing committee, the party leadership can't actually remove that member just from that committee—at least not until there is a new Congress," American University professor Andrew Ballard told Newsweek.
"President Trump's popularity has never been stronger than it is today, and his endorsement means more than perhaps any endorsement at any time," the former president's Save America PAC said in a statement.
An investigation into the Georgia congresswoman's social media activity found she had liked a comment that suggested shooting Nancy Pelosi in the head.
The House Republican leader has criticized the president's plans for COVID-19 and immigration reform.
On Twitter, people responded negatively to a response McCarthy gave to a question in an interview airing Sunday.
Cheney was just one of 10 Republican members of the House in favor of former President Donald Trump's impeachment.
The U.S. House plans to fast-track the next coronavirus relief package, aiming to have the proposal up for a vote in early February, after Democratic leaders rearranged the schedule to make way for President Joe Biden's $1.9 trillion priority proposal.