The passenger is also heard insulting supporters of Democratic candidate Hillary Clinton , branding them 'b****es'.
The incident occurred on board the flight from Atlanta, Georgia, to Allentown, in Pennsylvania on November 22.
In the 45-second clip, the man is seen shouting: "Donald Trump, baby!"
He adds: "That's right. This man knows what's up. We got some Hillary b****es on here. C'mon baby, Trump!"
On Saturday, Delta apologised for the incident and said the passenger should not have been allowed to continue on the flight.
Delta Chief Executive Ed Bastian added that, had staff witnessed firsthand what was shown in the video, there is no question the passenger would have been removed from the aircraft.
"He will never again be allowed on a Delta plane," Bastian said in an internal memo to employees posted on Delta's website on Monday, Mirror uk reports.
WATCH: Trump supporter calls women "Hillary Bitches" on flight to Allentown. Spoke to Emma Baum who shot video, @Delta did nothing. pic.twitter.com/2ZBjNCurNM— Yashar (@yashar) November 25, 2016