Gandhi, 29, is a member of the Bharatiya Janata Party in spite of being descended from two Congress Party prime ministers, Nehru and Indira Gandhi, Nehru's daughter. The BJP was originally a Hindu nationalist party but has been distancing itself from Gandhi since his alleged remarks were made public.
He is now a candidate for the parliamentary seat formerly held by his mother, Maneka Gandhi, in Uttar Pradesh, the Indo-Asian News Service said.
Gandhi asked why, if he had made the alleged remarks at an election rally on March 4, the video was not released for more than a week.
"I am a Gandhi, a Hindu and an Indian in equal measure and my attempt was not to spread ill-feeling against any community and I stand by all religion and all faiths," Gandhi said. "I am a victim of a political conspiracy ... Those are not my words, that is not my voice. I want to ask the nation to stand by me."
Gandhi said that Hindus in Uttar Pradesh are under attack.