The plane veered off the airport's main runway around 11:10 a.m. and skidded through a chain-link fence and perilously close to the edge of Flushing Bay. Photos showed the nose of the plane resting on a snowy berm that separated the runway from the icy waters.
There were 131 people, including crew members, on the plane. Six people had non-life-threatening injuries, and at least two were taken to the hospital, according to the Port Authority of New York and New Jersey, which runs the airport.
Donnell was unharmed and was posting photos and videos of the incident on his Instagram page.
"I'm blessed to be safe and sound after our plane skidded off the runway at LaGuardia airport today," Donnell said in a statement released by the team. "I feel fine physically and hopefully all the other passengers did not have any significant injuries. We were all shocked and alarmed when the plane started to skid, but most importantly, as far as I know, all of the passengers and flight crew were able to exit the plane safely."
The airport was closed shortly thereafter to incoming flights because the plane was leaking fuel, said Joe Pentangelo, a Port Authority spokesman.
"If we wouldn't have hit the snowbank, we'd be in the water right now," Charles Runels, a passenger from Atlanta, told reporters
The National Transportation Safety Board is sending an investigator to the scene to secure the plane's flight data and cockpit voice recorders and to document damage to the plane and other evidence, said spokeswoman Kelly Nantel.
Donnell was flying in from Atlanta to sign his 2015 contract tender. As an exclusive-rights free agent -- a player who has fewer than three years of NFL service but whose contract has expired -- Donnell is bound to his team for another year as long as it decides to tender him.
The Giants did so Wednesday, and Donnell will play 2015 on a one-year, $585,000 contract. He was third on the team with 63 receptions and 623 receiving yards and second on the team with six touchdown catches in 2014. He projects to be the Giants' starting tight end again in 2015.
The Associated Press contributed to this report.