After the pirates were shot, navy personnel sailed to the lifeboat and released Capt Phillips, whom they found tied up inside.
The ship's owner, Maersk Line Ltd, has also praised the captain's behaviour.
Chief executive John Reinhart said Capt Phillips had told him: "I'm just the byline, the heroes are the Navy Seals who brought me home."
In Eyl, a pirate stronghold on the Somali coast, one pirate chief reportedly threatened revenge against Americans.
"We will intensify our attacks even reaching very far away from Somalia waters, and next time we get American citizens... they [should] expect no mercy from us," Abdi Garad told the Associated Press by telephone.
Four French citizens, including a three-year-old boy, were freed aboard a yacht by French troops on Friday.
The yacht's owner, Florent Lemacon, was killed during the operation by French special forces, along with two pirates.
On Saturday, pirates hijacked a tugboat in the Gulf of Aden. The Buccaneer has 16 crew members on board, 10 of them Italians.
REACTION TO CAPT PHILLIPS' RESCUE
The sniper operation Sunday, with pirate guns aimed at Phillips, was a daring, high-stakes gambit, and it could have easily gone awry. If it had, the fallout would have probably landed hardest on Obama.
The result - a dramatic and successful rescue operation by US Special Operations forces - left Obama with an early victory that could help build confidence in his ability to direct military actions abroad.
A spate of attacks on ships off Somalia and the rescue Sunday of an American captain held hostage by pirates have reinvigorated a long-simmering debate over whether the crews of commercial vessels should be armed.
For Phillips captaining a ship turned out to mean more than safely manoeuvering around shallow shoals or managing a crew. It meant taking on the pirates who dared for the first time in centuries to attack an American-flagged ship.
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