WASHINGTON — Speaker Paul D. Ryan pledged to “run through the tape” and finish out his term when he announced last month that he would retire from Congress. But with Republicans in revolt on both his right and his left, Mr. Ryan is increasingly facing questions about whether he can manage to stumble across the finish line.

Mr. Ryan’s fractious conference has always been a management headache for him. But now moderates and conservatives are engaged in open warfare over one of the toughest issues before Congress — immigration — and the speaker is stuck in the middle, with no clear path forward. His colleagues know he will be gone in eight months, which is diminishing his leverage.

Mr. Ryan, of Wisconsin, also finds himself caught between emboldened conservatives taking aim at the Justice Department’s Russia investigation and more moderate voices, including top Republican officials at the department itself, desperate for him to intervene to stop them.

“Look, the members drafted me into this job because of who I am and what I stand for,” Mr. Ryan told reporters on Tuesday, as he insisted that he intended to stay in the job. “I think members very much agree that what we should be doing is completing our agenda and our work.”