Villavicencio, who worked at a pizzeria an hour away by auto in Queens, produced a city identification card, but the official told him he wanted to see a state driver's license, Chica said.
Villavicencio's wife, Sandra Chica, is an American citizen and their daughters, ages 2 and 3, were born in the U.S.
Though he will remain in detention at the Hudson County Correctional Facility in Kearny, New Jersey - in the custody of ICE - the stay will allow Villavicencio to work on obtaining permanent USA residency and hopefully be reunited with his family, legal aid said.
Make The Road says it's providing legal help to Villavicencio's family.
"Every time he called, he's just crying asking me about them", his wife said. "I confirmed with her that New York State, working with our partners, has secured pro-bono legal counsel to represent Mr. Villavicencio", he added in a separate tweet.
Villavicencio was detained by ICE officials on June 1 when he tried delivering pizza to the Fort Hamilton Army Base, an area he has delivered pizza to multiple times. But the man said, 'I don't care.' He said I need to keep waiting and he called ICE'.
Cuomo said on Thursday that he spoke by phone with Chica to express his "deep frustration with the federal government's assault on New York's immigrant families".
Villavicencio, 35, lives with his wife and two young daughters on Long Island.
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"My daughters need me", he added.
The couple married five years ago and this year Villavicencio applied for his green card.
"As such, his voluntary departure order became a final order of removal", she said.
Immigrations and Customs Enforcement agents arrested him when guards at the base questioned his New York City-issued ID.
"I haven't told them anything".
But he remains locked up at an Immigration and Customs Enforcement detention center in Kearny, N.J., pending arguments over his motion for a writ of habeas corpus and a temporary injunction barring his deportation. "I'm listening", New York City Councilman Justin Brannan said.
Redmond Haskins, a spokesman for The Legal Aid Society, a not-for-profit organization that provides free services to clients who can not afford them, said Villavicencio has no criminal record.
He said he showed his New York State identification card at the security gate and was allowed inside.