Welsh, a former coach at Iona and Providence, never worked a game at Hofstra after signing a five-year contract for $3 million to replace Tom Pecora, who left to coach Fordham in March.
"The university accepted the resignation in the best interests of the university and of the men's basketball program," Hofstra spokesman Stephen Gorchov said in a statement.
There was no immediate word from the school on a successor.
The 49-year-old Welsh was arrested Friday morning after Nassau County police found him behind the wheel of his 2006 Lexus, stopped at a green light at an intersection in Levittown at 1 a.m. He pleaded not guilty at his arraignment and is due back in district court Tuesday.
Welsh's voicemail box was full. His attorney declined to comment.
Welsh told Newsday in Saturday's editions that he wanted to "express the deepest regrets and apologies over the incident."
Welsh took over a program Pecora led to a 155-126 record in nine seasons, including three straight NIT appearances from 2005-07.
Welsh was fired by Providence in 2008 after 10 seasons, and spent the past two years as a basketball commentator for ESPN.
At Providence, he had an overall record of 160-143, bu was 1-9 in the Big East tournament. The Friars reached the NCAA tournament under Welsh in 2001 and 2004, losing in the first round both times.
His best season at Providence was 2003-04 when the Friars finished third in the Big East and were ranked as high as No. 12 in the country. Before Providence he spent three seasons at Iona, where his record was 70-22 with one NCAA and two NIT appearances.
Hofstra joined the Colonial Athletic Association in 2001. While it has had success on the court, the Pride have not been able to reach the NCAA tournament as a member.
Welsh is accused of aggravated driving while intoxicated, which means prosecutors contend he had a blood-alcohol reading over 0.18, more than double the legal limit of 0.08.
His arrest occurred in a county where the district attorney has built a reputation for zero tolerance on drunken driving. During her first term, Kathleen Rice obtained a rare murder conviction against a drunken driver who crashed head-on into a wedding limousine, killing a 7-year-old flower girl and the chauffer.
Marge Lee, an anti-DWI advocate whose daughter was killed in a drunken driving crash, said high-profile arrests like Welsh's help to focus attention on the issue.
"Unfortunately, everyday drunken driving arrests are not always in the paper," she said. "If every single arrest made this kind of headline, it might help bring an end to the problem. A better headline would be `drunken driving is over."'
Associated Press Basketball Writer Jim O'Connell contributed to this report.