"He was able to get from her by threat the address of the man," Medina County Sheriff Randy Brown said.
Garibay then drove about 160 miles northwest from Corpus Christi, which is along the southern Texas Gulf Coast, to a home near the town of Hondo where he fatally shot 35-year-old Keith Martin, Brown said. Martin's father and brother were at the home during the shooting, the sheriff said.
Garibay fled the scene and led deputies on an 18-mile chase west to the town of Sabinal before the officers forced him off the road, Brown said.
He said Garibay resisted arrest and was brought under control with a stun gun. Brown said deputies believed Garibay might have tried to overdose on medication.
Garibay was taken to a hospital in Hondo to be evaluated by doctors. He remained in custody at the hospital Thursday evening, Brown said.
Authorities in Medina County initially charged him with evading arrest, but Brown said, "There will be a murder charge coming soon."
Corpus Christi police have charged Garibay with assault and unlawful restraint and issued a warrant for his arrest on those counts, Brown said. Garibay's wife was treated for minor injuries, according to Corpus Christi police in a statement.
Officials with U.S. Customs and Border Protection did not immediately return phone calls seeking comment Thursday evening.