Yogvedant Poddar, 21, was removed from the Jet Airways flight after a passenger spotted him typing "Terrorist on flight, I destroy women's hearts" into his phone.
The caption accompanied a photo he had taken of himself which showed half of his face covered with a handkerchief, Kolkata deputy police commissioner Avvaru Ravindranath told AFP.
Irish national Benjamin Plackett was sitting behind Poddar on the Monday Kolkata to Mumbai service flight and alerted the cabin crew as the plane was being prepared for take-off.
Poddar was taken off the flight and detained for almost ten hours before being let go without charge, Ravindranath said.
"We questioned him extensively. We found no terror link. It was a message that was meant exclusively for his friends and was not intended to threaten or hurt anyone," said the police official.
Jet Airways said in a statement that a passenger had been "deplaned for security reasons".
"The aircraft departed for its destination with a revised flight timing" after the necessary checks had been carried out, the airline added.