I think this might be The Disappearance Trilogy by John Birmingham.
There are three books: Without Warning (2008), After America (2010), Angels of Vengeance (2011).
On March 14, 2003, a wave of energy washes over North America, wiping
out most of the populations of Canada, the United States, Mexico, and
Cuba. The source is unknown, but it appears that Seattle, Western
Canada, Alaska, and Hawaii are unaffected by the disaster.
This wave of energy is referred to as the "Wave" throughout the series.
In Without Warning, James Kipper (formerly a Seattle City Council Member) is eventually appointed President of the USA after the former President and almost his entire possible lineage of successors are wiped out. The only surviving Cabinet member is not a natural born US citizen and therefore ineligible.
In After America, President Kipper goes to New York in 2007 wanting to rebuild. Of course, chaos reigns.
After a meeting with foreign dignitaries in Seattle, Kipper and Culver
head to New York City, where salvage crews are working to clear the
streets of debris, the Wave having made its victims disappear, so
crew-less vehicles and aircraft crashed as a result. Due to the lack
of people to maintain New York City as well as cities all over the
contiguous United States, fires burned unchecked and New York City is
now flooded in some areas. The U.S. military has also had to step in
alongside the New York Militia to fight droves of pirates, looters,
terrorists and organized gangs who have moved into the U.S. East Coast
from South America, Africa, and Eastern Europe.
President Kipper holds a press meeting at Castle Clinton in Battery Park (not island) when the area is hit with Katyusha rockets from Muslim fighters on Ellis Island in an assassination attempt. So he was at Battery Park, though not living there.