Mr. Babeu made national headlines for his tough stance on illegal immigration. He endorsed Mr. Romney in October and is running for Congress in Arizona’s Fourth District. The news of Mr. Babeu’s decision to step down from the Romney campaign was first reported by The Arizona Republic. On Thursday, The Phoenix New Times reported the allegations of a 34-year-old Mexican man whom it identified only as “Jose.” He said that when he refused to sign an agreement that he would never mention his relationship with Mr. Babeu, which he said lasted for years, the sheriff and his lawyer tried to intimidate him with threats of deportation. In a press conference on Saturday, Mr. Babeu said that he had had a personal relationship with Jose, but denied that he had threatened to deport him. The accusations, he said, “are absolutely, completely false — except for the issues that refer to me as being gay. Because that’s the truth. I am gay.” Even though Mr. Babeu was a minor figure in the Romney campaign, the accusations came at a less than ideal time. With Arizona and Michigan holding primaries at the end of the month, Mr. Romney is struggling to shore up social conservatives’ support and to fend off Rick Santorum, who is surging after winning contests in Colorado, Missouri and Minnesota. “Sheriff Babeu has stepped down from his volunteer position with the campaign so he can focus on the allegations against him,” Andrea Saul, a campaign spokeswoman, said in an e-mail statement. “We support his decision.” The press conference that Mr. Babeu held on Saturday largely turned into a call to look beyond his sexual orientation. He defined himself as an American, and appealed for voters and the public to view him through the lens of his military service and time in law enforcement. “I want to be judged on my service: 20 years in the military, two deployments — including one in Iraq — a police officer who has responded to thousands of calls for help, and a sheriff who has cut response times while reducing my own budget,” he said in a statement. Jose also told The Phoenix New Times that Mr. Babeu e-mailed explicit photographs of himself to an anonymous man whom he met on a Web site called Adam4Adam.