House Minority Leader Kevin McCarthy Wednesday made his case to Republicans on why he should be the next House speaker, arguing that flipping the chamber under his leadership was "no small feat." 

"Over the past century, Republicans had flipped the House from Democrats just four times: 1946, 1952, 1994 and 2010. Only twice did that flip occur in four years or less," McCarthy, R-Calif., said in a letter to fellow Republicans. 

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"This was the most expensive and arguably the most competitive House midterm in America’s history," the minority leader continued. "Yet in a hard-fought contest, our message and our candidates prevailed, winning key seats across the country, some in districts that President Biden carried just two years ago by double digits…. This is no small feat." 

McCarthy's missive comes after an election night that fell short of most Republicans' hopes and expectations. 

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A GOP majority would likely be very thin, reducing McCarthy's margin of error in a House speaker race, as well as when governing in a majority – just a few members defecting on a vote could stall key legislation. 

But McCarthy argued he is up for the task of leading the House on behalf of Republicans. 

"I trust you know that earning the majority is only the beginning. Now, we will be measured by what we do with our majority. Now, the real work begins," he said. "That is why I am running to serve as Speaker of the People’s House and humbly ask for your support."

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"I will be a listener every bit as much as a Speaker, striving to build consensus from the bottom-up rather than commanding the agenda from the top-down," McCarthy added. "That means putting the right people in the right spots and harnessing the power of our entire conference."

It's not yet clear if McCarthy will face any serious challenge for speaker. House Minority Whip Steve Scalise, R-La., said on Fox News Wednesday that he supports McCarthy and will not challenge him for the role. 

Also, the House Freedom Caucus, which is poised to try to use its leverage in a thin majority to influence House GOP policy, has not yet begun to coalesce around any possible McCarthy challenger.

Fox News' Matthew Smoot contributed to this report.