In his role as a television pundit for ITV, Keane called for a change of mentality among both the players and the fans after Trapattoni's men were serenaded from the stands at the PGE Arena in Gdansk as Thursday's 4-0 thumping by Spain drew to a close.
But the 73-year-old said on Friday night: "Roy Keane was a very great player. We know Ireland, when we played against him, was a strong team. This generation was very hard.
"It is not my habit to be critical of other colleagues as coaches or managers, but what has he done after he finished playing?
"The players were strong in his time, playing at Manchester United, playing at Celtic, playing everywhere. They were a strong team.
"Roy Keane was a great player, a very great player. He obtained very great results as a player. Now he is a coach he should just stick to focusing on getting results as a coach."
Keane, who famously returned early from the 2002 World Cup finals in the Far East in protest at Ireland's preparations, was annoyed by the scenes at the end of the game.
He told ITV1: "It's a reality check for a lot of the Irish players. I think a lot of them think they are top players, and it goes to show that they are so far behind a lot of these players.
"I think the players and even the supporters, they all have to change their mentality. It's just nonsense from players speaking after the games about how great the supporters are.
"Listen, the supporters want to see the team doing a lot better and not giving daft goals away like that. I'm not too happy with all that nonsense.
"To praise the supporters for sake of it...let's change that attitude towards Irish supporters.
"They want to see the team winning - let's not kid ourselves, we are a small country, we are up against it, but let's not just go along for the sing-song every now and again."
Thousands of Ireland fans have flocked to Poland to witness the Republic's first trip to a major tournament in a decade, earning not only there praise of the Irish camp, but also Spain boss Vicente Del Bosque after Thursday night's game.
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