Crew members have hailed Phillips for giving himself up to the pirates last Wednesday to save their lives. Today crew members gave a brief emtional news conference, praising Phillips' heroism and their own ability to stick together.
"This crew was lucky to be out of it with every one of us alive," said Capt. Shane Murphy, of Buzzards Bay, Mass., the Alabama's second in command. "We're not going to be that lucky again."
"And just for the record, we never had to fight to take our ship back. We never surrendered our ship," Murphy said.
Phillips refused the title of hero and said his Navy rescuers are "the real heroes."
Obama phoned Phillips on the Boxer and also called Phillips' wife and family at their home.
In a prepared statement, he hailed the heroism of the military and Phillips.
"I share the country's admiration for the bravery of Capt. Phillips and his selfless concern for his crew," Obama said. "His courage is a model for all Americans."
Capt. Joseph Murphy, whose son, Shane, took over the Maersk-Alabama after Phillips was taken off the boat, seconded that feeling.
"Our prayers have been answered on this Easter Sunday," Murphy said in a written statement. "I have made it clear throughout this terrible ordeal that my son and our family will forever be indebted to Capt. Phillips for his bravery. If not, for his incredible personal sacrifice ... this kidnapping and act of terror could have turned out much worse."
Murphy made an appeal for Obama to take the lead in the fight against pirates.
"America has to be in the forefront of this crisis. And it is a crisis," Murphy said.
On ABC News' "This Week," Adm. Thad Allen, the commandant of the U.S. Coast Guard, said the answer is not arming merchant ships to defend against pirates, as some have advocated.
"I think that's pretty problematic for reasons such as training, certification, how you apply standards," he said. "The discussions I've had with the private sector and the ship companies really don't favor that right now."
Kaj Larsen, a former Navy SEAL who has made documentaries on pirates in Indonesia and arms sales in Mogadishu, told ABC News the problem of piracy will not be easily solved.
"I'm very relieved that the hostage is safe," Larsen said. "That was a happy ending to what was obviously a trying ordeal for the American crew. At the same time I'm cautiously pessimistic, because I'm keenly aware that we're going to see more and more of this problem in the future.
"I don't think in this particular case unfortunately, you're going to see a deterrent effect," he said. "The sums that these pirates are making are just extraordinary. So the incentive is too great even if they lose a few of their foot soldiers in the process."
The Maersk Alabama was in Somali waters because it was carrying food aid to hungry people in Africa, including Somalia.
ABC News' Jim Sciutto, John Hendren, Jake Tapper and Jason Ryan contributed to this report, as did The Associated Press