John Fury - Tyson Fury's father - stood in the WBC heavyweight champion's position on the stare down and appeared to say something to Whyte's brother Dean before being moved away by Tyson.
Fury and Whyte then told their camps to step back and shook hands ahead of their fight on Saturday.
Speaking to BT Sport after the altercation, Whyte said: "His dad needs to relax, my man is 600 years old and stepping forward, trying not to let my guys on the stage! It's mine and Tyson's day, why are you getting involved?" He's an idiot."
"I told him, 'John calm down, you're an adult. You keep shouting and screaming like a child, relax'.
"It's our day. Why are you getting yourself involved, this is mine and your son's day. He's always trying to get in front of his son. I just said old man take it easy.
"John was trying to start something. He's stupid because it's a big fight. He's trying to rob 90,000 fans of a fight. John Fury is an idiot, he's a real idiot."
Fury refused to deliver any trash talk at Wednesday's press conference and said the fight does not need selling through bad blood.
"Listen, he looks ready, he's in shape, I'm in shape, it's going to be a good fight. I've just prevented a big ruckus on stage. People want a big ruckus, they want to see punching and screaming, but let the two boxers do a fight and enjoy it. It's not for entourages to get involved.
"We're here to do a fight. People are in for a real fight. We don't need to scream and shout. It was all good, very respectful. The man's done his job, he's obviously trained, so have I, end of."
Meanwhile, Whyte is prepared to adapt during the fight against a Fury who can switch from orthodox to southpaw and a heavy puncher to a slick boxer.
"This fight is one of those fights where you've got to keep an open mind. It's about being able to adapt and quickly.
"Fury says one thing and he does another thing. He might decide he wants to fight or box, so I have to keep adapting to whatever he does. I'm not going to envisage this or that."