Mr Campbell accused the shadow chancellor of following the Conservative Party with a similar direction and said Labour would end up in the “same place” after Brexit.
Labour’s Mr McDonnell rejected the argument and claimed Mr Campbell was trying to “divide” the British public with his interventions.
The debate culminated in a huge row on Peston on Sunday, and started as Mr Campbell accused Labour of ignoring the “cost” of Brexit to the United Kingdom.
“They’re both going to end up in the same place,” Mr Campbell said.
“They are basically saying whatever the cost, whatever the chaos, whatever the damage to the economy, whatever the Brexit impact to assessment is saying, a double-digit fall in growth in the north-east of England, they may have voted Leave but they are the ones hardest hit.
“We are still basically on the road out and it doesn’t matter what happens, that is going to happen.”
Peston asked whether Mr McDonnell agreed with the assessment by Mr Cambell as the Labour politician was shaking his head throughout his tirade.
Labour’s shadow chancellor said: “No look, Alastair, stop dividing us.”
No look, Alastair, stop dividing us
The pair began speaking over each other before Mr Campbell was asked to let Mr McDonnell answer the question.
Mr Campbell raged: “No, he has just accused me of being divisive.
“Party members want to see more leadership on this.”
Mr McDonnell replied: “That is the sort of macho-threatening politics that has divided this country.
“What we need is to have a rational, moderate debate about the pros and cons about our future and the relationships we need to exaggerate individual people or parties position doesn’t help that.
“What we are trying to do is to bring people together about building our future.”
Again Mr Campbell attempted to interrupt the Labour frontbencher, claiming he was saying “whatever happens we are out of Europe.”
The shadow chancellor snapped back at Mr Campbell one final time and insisted that he “accepted the result of the referendum”.
He said: “We respect the referendum result and we have to build a new relationship with Europe which will bring our people together again.
“I think that is doable because I think in a traditional British compromise, that is what people want.
“The compromise is tariff-free access to the single market, yes a customs union if necessary if we can agree an appropriate one. And then building upon a future Europe that is prosperous on the basis of trade between European countries.”
During his interview with Peston, the Labour shadow chancellor also outlined exactly what he was seeking in “a customs union” with the EU.
He said: “It’s the ability to influence trade deals that will go off in the future.
“The argument is that if we remain in the customs union, like Turkey, then you don’t have that same level of influence.
“So we are suggesting a customs union should be on the table for negotiations.
“What we are saying to the Government is don’t take that off the table because I think there is a constructive debate to be had about that.”