Police officials said it was a mystery to them why Officer Rodriguez made up his story.
"I never gave up the dream, even while being a cop," Officer Rodriguez, 37, said of his singing career.
The commissioner said the department was leaving it up to Officer Rodriguez to decide whether to stay with the force.
But Officer Rodriguez seems reluctant to leave the financial security.
The police said they began investigating the allegations of abuse against Officer Rodriguez late last year.
Officer Rodriguez, 23, has been with the department for three years, she said.
Family members said Officer Rodriguez's police work meant everything to him.
Officer Rodriguez has been charged with two counts of second-degree assault.
Officer Rodriguez opened the door to the stairwell with his gun drawn, the police said.
Officer Rodriguez discovered the counterfeit bills during a routine search.