The 143-metre Sea Diamond was operated by Louis Cruise Lines, part of a Cyprus-based tourism group.
Search for passengers
Greek navy divers were searching the vessel for signs of the missing passengers.
Palli Petralia, Greece's tourism minister, said she had spoken with the missing passenger's wife.
"Whoever is responsible for this will be held accountable in the strictest way," Petralia said.
"Greece is a major tourism destination and incidents like this must not be allowed to occur. Authorities handled the rescue very well."
No injuries were reported during Thursday's dramatic three-hour rescue, which involved Greece's military, commercial ships and the island's fishermen.
Many passengers were forced to climb down rope ladders to coast guard boats some 25 metres below.
The Sea Diamond struck rocks in a sea-filled crater formed by a massive volcano eruption 3,500 years ago.
"Whoever is responsible for this will be held accountable in the strictest way"
Palli Petralia, Greece Tourism Minister
Tourists gathered on overlooking cliffs to watch the rescue effort.
Louis Cruise Lines said there were 730 Americans, 112 Spaniards, 100 French and many other foreign nationals on board, including Germans, Britons and Australians, plus 390 crew.
Most of the rescued passengers arrived at Athens' main port of Piraeus on Friday on a chartered ferry and a Louis cruise ship.
Authorities on Santorini said they were working to contain a small oil spillage from the sunken ship.
The Sea Diamond's captain and three officers were being interviewed Friday by coast guard investigators who flew to Santorini.
In September 2000, 82 people drowned when the Express Samina ferry hit rocks and sank off the island of Paros, a disaster that led to a major improvement in Greek ferry safety standards.