Coming as a tourist 15 years ago, Mr. Hernández García stayed and built a business laying floor tiles, now with six employees. At a construction site in 2010, he had an argument with a worker from another crew who had trod on a freshly laid floor. The man called the police and Mr. Hernández García was arrested on an assault charge. His fingerprint check drew the attention of immigration agents.

He insisted on a trial, and was acquitted after witnesses confirmed that there had been no attack. “I’ve had all these problems because one person spoke falsely,” Mr. Hernández García said.

He and his wife, Ana, 40, who is Mexican, applied years ago for green cards through her father, an American citizen. Their applications are buried in processing backlogs. They have a Mexican daughter they brought here as a baby; another daughter, 9, was born in the United States. But an immigration judge who heard his deportation case was not convinced that it would be a hardship for his family if Mr. Hernández García was deported.

He spent New Year’s Eve in a chilly basement room at the First Unitarian Church with his family, singing karaoke to melancholy mariachi songs. The congregation has agreed to shelter him, since immigration agents generally refrain from entering churches to make arrests. This week, immigration prosecutors in Denver declined to cancel his deportation order, so he will present a new application to have it temporarily postponed.

“I want the citizens of Denver to know,” he said, “that we are not criminals.” Except for his deportation order, Mr. Hernández García has all the qualifications for a work permit under Mr. Obama’s new program.

Started in 2008, Secure Communities rapidly expanded under Mr. Obama to cover the entire country. Under the program, fingerprints of anyone booked by local or state police officers were shared with immigration agents. The agents could then issue orders, known as detainers, to the police to hold people suspected of being in the country illegally for up to 48 hours after release from local custody, giving agents time to detain them for deportation.

Official statistics show that agents modified their operations over time to deport more convicted criminals. But the government continued to expel immigrants following traffic stops or misdemeanor arrests, or after charges that never resulted in convictions.