When Wright accused him of not being happy for the England team, Keane replied: "I don't mind you being happy, but you were planning the final, where the parades were".
"You're not going to get a shout in a World Cup semi-final".
The question divided the pundits after the Three Lions fell to a 2-1 extra-time defeat against Croatia.
Ahead of the game, Keane was already drawing the ire of English supporters on the "It's Coming Home" bandwagon.
Croatia into the World Cup Final!
Did we expect to be in this position? "We deserved the victory", said match-winner Mandzukic. Croatia have hardened warriors whose understanding of big matches at times came to the fore.
Wright, for his part, defend his right to get excited in ITV's coverage and took revenge by mocking Keane's accent.
That caution proved to well founded as Mario Mandzukic's extra-time victor fired Croatia into Sunday's final against France as England's expense in a tense semi-final in Moscow, with Keane biding his time before revisiting his previous discussion with Wright.
Two fiery players during their own playing days as prominent figures of arch rivals Manchester United (Keane) and Arsenal (Wright) in the 1990s, the duo seem to have carried on their rivalry into the TV studio in what social media users have dubbed "one of the TV moments of the year".
"Before the game, all the talk was about the final".
Just some very sad English soccer players on the ground
The rising hope only makes the dip of despair even more chastening. "No way will they have such an easy route next time". And then, just when England thought old fears could be dismissed and new chapters of success written, it happened again.
However, he was interrupted by Wright, who insisted: "We wasn't talking about the final, we were just having a laugh with ya".
Keane, never one to let a potential dust-up pass, wasn't letting Wright forget that 1966 world champions England had been sent packing.
The quartet of former footballers were last together earlier in the tournament and viewers got a real kick out of Keane telling his colleagues to cool the jets after they beat Panama 6-1.
Gareth Southgate's men still have one more game to play as they will take on Belgium in the third-place play-off on Saturday. "We were happy! Why can't we get excited?"
Coach Southgate forced England to face up to Iceland debacle
Asked what is was all about, a slightly bemused Southgate said: "That's exactly what I asked our fitness coach a well, 'What is that all about?'".
A fired up Keane started the debate as he suggested many England fans had lost track of reality as they got carried away with the excitement of their World Cup run, with Wright eager to snap back at the Irishman in typically jovial fashion.