More than 1,600 people were onboard the Greek-flagged ship, run by Louis Cruise Lines, after it hit a volcanic reef close to Santorini's main port.
Greek tourism minister Fanny Palli Petralia confirmed that the 45-year-old man and his 16-year-old daughter were missing.
She said the missing man's wife and another of the couple's children were rescued and had spoken with officials about how the cabin they were staying in had filled with water.
Dramatic TV pictures this afternoon showed the 1986-built vessel disappear under the water as life rafts floated nearby.
Rescued tourists taken ashore yesterday have spoken of the panic that swept through the ship.
Passenger Katie Sumner told BBC News that there was no alarm sounded as water began to rush onboard.
"We had no time to get ready or anything," she said. "We were just left where we were."
The Sea Diamond had left the port of Piraeus on Monday and was carrying a passenger-load dominated by American and German holidaymakers.
Officials say that 1,195 passengers and 391 crew were on board.
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