When President Trump called Representative Max Miller, his former loyal aide, to warn him the domestic abuse allegations he was facing would make it tough for him to win re-election, Republican officials at the White House and in Ohio hoped it would be enough to get the embattled congressman to drop out.

But Mr. Trump stopped short of asking Mr. Miller to step aside, and the congressman made it clear he had no intention of doing so, according to two people familiar with the call who spoke about the confidential conversation on the condition of anonymity.

The president, who endorsed Mr. Miller in May, has yet to publicly withdraw his support, leaving other Republicans reluctant to do so themselves.

The exchange, which was reported earlier by Axios, has left the G.O.P. in a bind as officials scramble to figure out how to protect a House seat that the party cannot afford to lose in the midterm elections, with a headstrong candidate and a fast-approaching deadline to get him out of the race.

If Mr. Miller were to drop out by the end of the day on Tuesday, there would be a special primary election to replace him on the ballot. Otherwise, Ohio law suggests he would need to resign by Friday to trigger a meeting of state party chairs, who would have to choose a new candidate by Monday.

The tight time frame has transformed what up until recently had been a family feud between two of the state’s most powerful families into a political crisis that has forced Republican officials to consider picking sides.

The drama, which has played out for months in court and in dueling media appearances and leaks, has pitted Mr. Miller, 37 against his ex-wife, Emily Moreno, 32, who has accused him of abusing her and their toddler daughter. Mr. Miller has denied the claims and said that Ms. Moreno, the daughter of Senator Bernie Moreno, also an Ohio Republican, is mentally unstable, accusing her of stalking and lying about him and endangering their daughter.

The concern from the highest levels of the Republican Party is not personal but political. The approaching ballot deadline has focused fresh attention on Mr. Miller’s case, including court filings that detail Ms. Moreno’s charges that he held a gun to her head, scalded her with water and fractured their daughter’s collarbone.

Mr. Moreno, who had initially made little in the way of public statements about the matter, issued a blistering one on Sunday calling his former son-in-law unfit for public office. But few other Republican leaders have openly criticized the congressman over the accusations, even as they privately worry that the scandal could put Ohio’s Seventh Congressional District at risk.

The district, which stretches from the Cleveland suburbs to Amish country, is solidly Republican. But with anti-Trump sentiment sweeping the country, it is not a lock, especially given that Mr. Miller’s Democratic challenger, Brian Poindexter, is a union ironworker who could appeal to working-class voters.

White House officials have been strategizing on how to most effectively prod Mr. Miller to withdraw. They have also explored possible replacements should Mr. Miller exit the race, according to a person familiar with the conversations.

Jane Timken, a state senator and former state party chair who unsuccessfully sought the G.O.P. nomination for Senate in 2022, has been floated as a possible alternate — though she has made clear that she does not want the job.

Mr. Miller, though, has insisted in private and public that he can still win a third term, and angrily rejected any suggestion that he step aside. He broadcast a lengthy livestream over the weekend in which he appeared agitated as he denied the allegations against him. On Tuesday, after having spoken with Mr. Trump, he announced that he would open an ethics investigation into himself “to clear my name from these horrific attacks on myself and family.”

Not long afterward, the House Ethics Committee said in a statement that it was “reviewing allegations that Representative Max Miller may have engaged in domestic violence and abuse or illegal drug use.”

A long paper trail of court filings, including for dueling restraining orders, as well as a file that Mr. Miller released on Sunday in an apparent effort to clear his name, paint the picture of an increasingly ugly family dispute in which each side has gone to great lengths to make the other look as violent, unstable and dishonest as possible.

The back-and-forth is full of tragic details, including a story that Mr. Moreno told on Tuesday at the Capitol, in which he said he had once picked up his granddaughter at a police station and that Mr. Miller had refused to hand over her favorite stuffed blue bunny.

Also on Tuesday, Ms. Moreno’s lawyers pointed to a picture of the couple’s toddler that had been included in the file Mr. Miller posted on Sunday and accused him of publishing an image that could be considered child sexual abuse material, noting that it had been viewed at least 204,000 times before it was removed.

A lawyer for Mr. Miller said the inclusion of the photograph was a mistake and that he had taken it down, according to Punchbowl News.

On the ground in Mr. Miller’s district this week, there was little sign of the chaos sweeping the race and alarming Republicans at the highest levels. Mr. Miller showed up on Monday at a county fair, where he campaigned before a hog show and among constituents who mostly said they had not heard about the week’s developments.

Local officials also stayed mostly silent about the race. Three of the four Republican county chairs in Mr. Miller’s district did not respond to questions on Monday about whether they supported his candidacy. Matt Miller, the chairman of Ashland County and the mayor of Ashland, said he “did not have a final opinion on the matter.”

Gov. Mike DeWine of Ohio, a Republican, told a reporter for Signal Ohio on Tuesday: “I’m not going to weigh in on it. These are very serious allegations that he vehemently denies.”

Vivek Ramaswamy, another Republican who is in a competitive race for governor, did not respond to inquiries about Max Miller’s situation. And Senator Jon Husted, who is facing his own tough race to keep his seat, told Newsmax on Tuesday only that he hoped “the president’s intervention can help move things along.”

Behind closed doors, however, some local officials were working to figure out how to compel Mr. Miller to step aside, according to two people briefed on the conversations, and what would happen if he did. They had come to the conclusion that only Mr. Trump himself had a chance of changing Mr. Miller’s mind.

Former Representative Jim Renacci, a Republican and former small-town mayor who represented a House district that covered much of the same area from 2011 to 2019, said he had been fielding calls about stepping in to seek the seat.

“I am convinced of this: The only person that would have a chance, and might not even have a chance, is me,” he said from a bench in downtown Wadsworth on Monday night, citing his name recognition in the district and the difficulties of running a potentially short race. “And even when I think of it, it would be a daunting challenge.”

Mr. Renacci also stopped short of saying that Mr. Miller should withdraw.

At the Medina County Fair, where Mr. Miller appeared on Monday, the Republican Party booth was advertising his picture next to a sign that said “RE-ELECT.”

Volunteers declined to discuss his campaign, saying only that they were praying for him.

Across the aisle, at the Democratic Party booth, volunteers were registering voters.

Tyler Pager contributed reporting from Washington.