Yong Yow, the ICE spokeswoman, said that in March 2010 Villavicencio was granted voluntary departure by an immigration judge but failed to depart by July, as ordered. If the court grants a stay, he can remain in the country as he tries to fight the removal.
An active ICE warrant for Villavicencio was on file because he had not complied with an immigration judge's voluntary departure order in 2010.
The ruling is "a reminder that the judiciary can still serve as a powerful check when other branches of government make hasty, cruel and reckless decisions", Legal Aid's supervising immigration attorney told the Associated Press.
Villavicencio, who applied for a green card earlier this year, was born in Ecuador and lives in Hempstead, Long Island with his wife and two young daughters. Villavicencio, who worked at a pizzeria an hour away by vehicle in Queens, produced a city identification card, but the official told him he wanted to see a state driver's license, Chica said.
Wife, Villavicencio, Sandra Chica (Chica Sandra), said he went to deliver pizza to Fort Hamilton last week.More news: Ariana Grande gets engaged to beau after just one month of dating
The Legal Aid Society filed another lawsuit claiming Villavicencio was the victim of racial profiling and the detention violated his civil rights.
When Villavicencio arrived at Fort Hamilton, guards requested identification and he produced a city identification card.When one guard said the card wasn't valid and that Villavicencio needed a driver's license, he replied that he didn't have one.
Villavicencio entered the country illegally in 2008, reports the Times, and he has no criminal record. The governor declared his "deep frustration with the federal government's assault on New York's immigrant families" and called Villavicencio's arrest "an outrageous affront to our NY values".
Immigration and Customs Enforcement officials said the military police officers at the base then detained Villavicencio and turned him over to federal immigration officials.
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 US residency and hopefully be reunited with his family, legal aid said.
"There is absolutely no legitimate reason to proceed with an expedited removal, and to do so would be inhumane", Cuomo wrote.