The incumbent Republican is in a tough race against her Trump-backed opponent.
The petition has gained hundreds of thousands of signatures in the days since the Supreme Court overturned Roe v. Wade.
The Republican congressman described Cheney as "the only serious person" in a GOP midterm debate.
"I think the church played a huge role in how we got to where we are," the Republican congressman from Illinois said.
The former president called the Republican incumbent "terrible," saying she is "despised by almost everyone" on Saturday.
Polls suggest the Republican incumbent, who voted for Trump's impeachment, is headed to a runoff.
The congressman voted to impeach Trump after the Capitol riot but voted with him more than 90 percent of the time during his term.
A Trump spokesperson dismissed the Wyoming Republican's criticism, saying Cheney "has nothing but contempt for the people."
"It is contrary to everything conservatives believe, to embrace a personality cult. And yet that is what so many in my party are doing today," she said.
Liz Cheney, like Brian Kemp and Brad Raffensperger in Georgia, continues to be a thorn in the ex-president's side.
The scant polling out of the western state shows the incumbent Republican trailing her top challenger by double digits.
"If anybody tears up our Constitution...as what happened on January 6, I'm going to try to hold them accountable," the GOP congressman told Newsweek.
The Alaskan Republican previously beat the odds to win reelection as a write-in candidate in 2010.
"We could have lost our democracy that day," Congressman Tom Rice said during a GOP primary debate in South Carolina on Thursday.
"I think I have an obligation to support the nominee of my party, and I will," the top Senate Republican said.
The South Carolina congressman described the former president as a "would-be tyrant."
"That battle over the future of the Republican Party is playing out within our borders," the Casper Star-Tribune's editorial board wrote.
"January 6th was an unconstitutional attempt led by the president of the United States to overturn an American election and reinstall himself in power illegitimately," he said of Donald Trump.
"The Republican Party will not get better unless you stand up," the congressman said in a message to fellow GOP lawmakers on Saturday.
The GOP House Minority Leader has helped five anti-Trump Republicans raise about $100,000 each in the first half of 2021.
"There isn't going to be an impeachment, but I think [the Democrats] have a good chance of having a very bad election next year," the Senate minority leader said.
The former senior legal adviser to Donald Trump criticized Joe Biden's approach to the southern border amid the COVID-19 pandemic.
Some GOP lawmakers continue to push baseless conspiracy theories about the pro-Trump assault on the Capitol.
A poll released this week showed that two-thirds of Republican voters want Trump to run again in 2024.
"The American people deserve the truth," the Illinois Republican said.
All 10 of the House Republicans who voted to impeach Trump for his alleged role in the deadly January 6 Capitol riot have managed to out-fundraise their Trump-backed challengers.
House Speaker Pelosi said the embattled New York governor "should look inside his heart ... to see if he can govern effectively."
Just 14 percent of Republicans believe the former president should have been convicted by the Senate.
"I could see the pressure would become great for us to actually have to impeach Biden," California Representative Devin Nunes said Wednesday.
"This is not a Trump-first party. This is a country-first party," the Republican congressman said.