"Some remarks made by me in the heat and dust of the election campaign have caused dismay and consternation. I apologise unreservedly for them," Pal said in a written apology to the Trinamool Congress and the media.
"Whatever the provocation -- and there was an attempt at provocation -- those comments should never have been made," the letter said.
"In making them, I have let down my constituents and the people of Bengal, I have let down my party, the Trinamool Congress, and my political colleagues, I have let down my family, including my wife and children and my parents, and my friends," said Pal, who had remained incommunicado since the controversial video was aired in the media yesterday evening.
"I apologise to all of them, and particularly to all women in our society and to those in the media who highlighted the issue," the two-time Lok Sabha MP from Krishnagar said.
"I have no excuses to offer. It was a gross error of judgement and deeply insensitive. It should not have happened. And I assure you it will not happen again. Once more, a humble apology," he said.
Pal was asked by the Trinamool Congress leadership to explain his conduct in writing.
TMC supremo Mamata Banerjee has expressed shock and pain at the comments made by Pal.
Party general secretary Mukul Roy said that the explanation was part of a process which has been initiated.
"A process has been started, let's see where it goes," Roy said.