WASHINGTON — Endlessly divided, House Republicans pleaded with Rep. Paul Ryan on Friday to rescue them from their damaging leadership vacuum. But the Republicans’ 2012 vice presidential nominee showed little appetite for the prestigious yet thankless job of speaker of the House.

The Wisconsin Republican who chairs the tax-writing Ways and Means Committee – his dream job, he’s repeatedly declared – refused comment again and again as reporters chased him around the Capitol a day after Majority Leader Kevin McCarthy shocked his colleagues by withdrawing from the speaker’s race moments before the vote.

McCarthy’s abrupt decision came just two weeks after the current speaker, John Boehner of Ohio, announced his own plans to resign at month’s end.

That left Republicans in chaos, with a yawning void at the top of their leadership ladder even as they confront enormous fiscal challenges and budgetary deadlines that could threaten a government shutdown and unprecedented default in the months to come.

So Republican lawmakers, from Boehner and McCarthy on down, turned to Ryan, 45, the only figure in the House seen as having the stature, wide appeal and intelligence to lead Republicans out of the mess they’re in.

“He’d be an amazing speaker,” McCarthy declared to a bank of TV cameras after Republicans met behind closed doors to discuss their predicament. “But he’s got to decide.”

Said Rep. Lynn Westmoreland of Georgia, himself a potential candidate for the job: “He’s the only guy who can unite us right now.”

Not long after, Ryan rushed out of the Capitol, refusing to talk to reporters.

Ryan’s spokesman, Brendan Buck, said: “Chairman Ryan appreciates the support he’s getting from his colleagues but is still not running for speaker.”

Why not? Possible reasons include the presidential ambitions he may well still harbor. The speaker’s post requires a huge commitment of time and effort in corralling a party’s House members. It is not on anyone’s tactical roadmap to the White House.

But Republicans were determined to do what they could to get Ryan to reconsider. Rep. Darrell Issa of California said he carried Ryan’s gym bag for him Friday morning in an effort to persuade him to run, and Ryan even fielded a call from his presidential running mate, Mitt Romney.

Romney later issued an effusive statement declaring: “Paul has a driving passion to get America back on a path of growth and opportunity. With Paul, it’s not just words, it’s in his heart and soul.”

The clamor for Ryan dominated Republicans’ interest while Democrats watched with a mixture of fascination and trepidation, concerned about the challenges just ahead for Congress.

Several Republicans were quick to warn that despite Ryan’s popularity, he too could fall victim to the ferocious crosscurrents that felled Boehner and blocked McCarthy’s ascent.

“The same people who wanted to take down John Boehner, who wanted to take down Kevin McCarthy, are going to want to take down the next guy, too,” said Rep. Charlie Dent of Pennsylvania.

Indeed some in the hardline House Freedom Caucus, the faction of 30-plus conservatives responsible for causing much of the House’s disarray, were already registering their disapproval of Ryan. And some outside conservatives were pointing to his support for immigration legislation and the 2008 Wall Street bailout as disqualifying him for the speaker’s chair.

“I think he has the same problems” as Boehner and McCarthy, said Rep. Paul Gosar of Arizona, a Freedom Caucus member. Gosar pointed to Ryan’s alliance with McCarthy and former Majority Leader Eric Cantor – who once termed themselves the “Young Guns” – and said, “They’re definitely conjoined.”

One Republican close to Ryan said that the only scenario where a Ryan speakership was likely would be if he were to be selected by unanimous acclamation, as opposed to having to bargain with the Freedom Caucus for their support in the same manner that undid McCarthy. This Republican demanded anonymity to discuss private considerations.

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