The video purportedly shows Pacheco driving his car on the beach, which is prohibited as per local laws.
Denying that he abused the man, later identified as Melroy D'Souza, the former minister said D'Souza was a "Portuguese national and his political rival" who "chased" his car in a jeep.
However, in a social media post, D'Souza claimed that Pacheco threatened him and even tried to run his car over him.
In the video, Pacheco is purportedly seen sitting with his friends in a beach shack and abusing D'Souza who also questionned the conduct of the former minister.
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"I became irritated with Melroy D'Souza as he was chasing me in his jeep on the road and also followed me on the beach," Pacheco told reporters in Margao today.
Pacheco said D'Souza had also abused him.
"I asked him to stop and talk with me, but he just ran away saying catch me if you can," said the former minister.
Pacheco, who was an MLA of the Goa Vikas Party, had to resign as tourism minister in 2015 after he was convicted for slapping a government servant on duty and sentenced to one-year jail term.