A new bishop is coming to the Diocese of Camden.
Sullivan said at a news conference that he was "delighted" to be leading the 500,000-member diocese.
Galante, who will reach the mandatory retirement age of 75 in July, suffers from chronic kidney disease and began dialysis treatments in October 2011. He has had Type II diabetes since 2000 and asked the Vatican a year ago if he could retire due to poor health.
Galante told the Philadelphia Inquirer in April that he recognized there was "the possibility that there may come a time [before age 75] when I might not be able to continue."
The pope accepted Galante's resignation, the Vatican said.
Sullivan, 67, has been an auxiliary bishop in New York since June 2004 and vicar general of the archdiocese -- one of the largest Catholic dioceses in the country -- since 2005. Sullivan was born in New York City and was previously a priest at various parishes there.
He serves on the subcommittee for Asian affairs for the United States Conference of Catholic Bishops. He also speaks Spanish and has served on the conference's Campaign for Human Development.
At the news conference this morning, he spoke fluently and at length in Spanish and said the church must stay committed to the poor.
In 2009, Sullivan said the funeral Mass for the Rev. Richard John Neuhaus, a leading conservative Catholic writer and intellectual.
Galante has led the Diocese of Camden since April 2004. Before that, the Philadelphia native was bishop in Beaumont, Texas, then coadjutor bishop of Dallas.
In April 2008, Galante stunned the diocese with news that he would close or merge nearly half of the 124 parishes.
"Inaction is not an option," he said at the time. This April, he defended the consolidation, which left 70 parishes across Camden, Gloucester, Cumberland, Salem, Atlantic and Cape May counties.
"We couldn't keep bailing out parishes," costing the diocese "millions," he told the Inquirer.
Still, the reorganization was highly contentious and prompted calls for his resignation, threats of lawsuits and petitions to the Vatican to overturn his plans. The Vatican has rejected some dioceses' consolidation plans, but upheld Galante's actions.
Inquirer staff writer David O'Reilly contributed.