WASHINGTON, May 14 (Reuters) — Republicans in the U.S. House of Representatives elected Donald Trump's candidate Elise Stefanik to their leadership, succeeding Liz Cheney, who they ousted earlier this week for criticizing the former president's continued false claims of election fraud.

Stefanik, a 36-year-old congresswoman from New York state, rose to prominence with her fierce defense of Trump during his first impeachment trial, while Cheney angered her Republican colleagues by rejecting what she called Trump's "big lie" about his November loss to Democratic President Joe Biden.

The secret-ballot vote boosted Trump's dominance over the party even after it lost its majorities in the House and Senate, as well as the White House, during his single term in office. Trump is positioning himself to play a major role in next year's congressional elections and is also flirting with a 2024 White House run.

"He is a critical part of our Republican team," Stefanik said. "I believe that voters choose the leader of the Republican Party and President Trump is the leader that they look to."

Stefanik defeated Rep. Chip Roy, who entered the race to serve as chair of the House Republican Conference on Thursday night.

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House conservatives including Roy had complained that Stefanik's voting record was not conservative enough, including a vote against Trump's 2017 tax cuts, his main legislative accomplishment. But the party in the end lined up behind Trump, who blasted Roy's entry into the contest in a statement, hinting at backing a primary challenger to him next year.

A woman has held the No. 3 House Republican leadership job since 2013, as the party has tried to broaden its appeal to independent voters, including suburban women.

The secret ballot tally, according to a source in the room, was 134-46, with nine voting "present" and three votes cast for other candidates.

Conceding defeat, Roy, a Texas congressman, said he succeeded in delivering a message to leadership that it should have created a more open process for electing Cheney's replacement.

"Elise is our chair," Roy said after the vote. "Now we're going to get busy, pointing out how the Democratic Party is destroying America."

Cheney to fight on

Cheney has vowed to continue to fight to prevent Trump from winning the White House again in 2024.

Stefanik was a fierce defender of Trump during his two House impeachment trials. Despite her somewhat moderate voting record, she defeated the rock-solid conservative Roy.

Following the closed-door vote, Trump issued a statement saying, "Congratulations to Elise Stefanik for her Big and Overwhelming victory!"

In celebrating Stefanik's victory, House Majority Leader Kevin McCarthy tried to put the bruising leadership battle behind Republicans.

During remarks to reporters with Stefanik at his side, McCarthy instead launched an attack on Biden, blaming him for an immigration "crisis" at the southern border, gasoline shortages on the East Coast due a cyber-attack on a major pipeline and unrest in Israel.

Democrats hold a razor-thin majority in the House and a 50-50 split with Republicans in the Senate. Given those numbers, and a history of political parties doing well in mid-term congressional elections when the opposition holds the White House, Republicans are optimistic of taking over the House in the November 2022 elections.

Now free of her job as head of the Republican conference, Cheney, a lawmaker with impeccable conservative credentials and the daughter of former Vice President Dick Cheney, vows to steer the party away from a man she says is "pushing the lie" that his defeat in the 2020 election was the result of massive voter fraud.

Trump's claim was rejected by multiple courts, state election officials and his own administration. Cheney calls him "an ongoing threat" to U.S. democracy.

House Republican leaders, along with many in their rank and file, pushed Cheney out on Wednesday because they had concluded her battle with Trump was distracting from attacks on Democratic President Joe Biden and threatening their prospects in the 2022 congressional elections.

By dumping Cheney and electing Stefanik, Republicans demonstrated that adherence to strict conservative ideology is no longer of prime importance in the Trump-era.

The conservative Club for Growth, which rates members of Congress, gives Stefanik a lifetime score of just 35% for voting in line with its priorities, one of the worst among House Republicans, and well below Cheney's 65%.

Even more striking, Roy enjoys a 100% lifetime rating.

While House Republicans, cheered on by Trump and McCarthy, were eager to shun Cheney this week, they have not acted against Representatives Marjorie Taylor Greene or Matt Gaetz.

The House's Democratic leadership stripped Greene of her committee assignments for past incendiary remarks that included support for violence against Democrats, while Gaetz is the subject of a federal child sex-trafficking probe.

Both are close allies of Trump.

(Writing by Richard Cowan; Editing by Scott Malone, Peter Cooney, Steve Orlofsky and Nick Zieminski)