Tom Bradby described in a series of tweets how he dived in after the man but "couldn't get him out".
The father-of-three, who was in Rafina on a family holiday, said he believed the man had committed suicide.
The Rafina Port Authority in Greece confirmed a man had died after his car went off the wall.
A spokeswoman said the incident happened at about 16:30 local time, but would give no further details.
Mr Bradby, who is ITV's political editor, said he was "shaking with shock and anger" after complaining that officials nearby did not do more to help.
He claimed that of the nine or 10 officials on the quay, one went in to try to get the driver out as the car sank, and "the rest watched".
Bradby, 48, has been announced as the new host of ITV's News at Ten.
He was previously a royal correspondent, and secured the first official interview with the Duke and Duchess of Cambridge after their engagement.
An ITV journalist has described how he tried in vain to save the life of an elderly man who drove his car off a harbour wall in Greece.