Veron has apologised to the people of Argentina following his side's failure to return home with the World Cup - and described himself as "devastated".
Veron said: "Both myself and the team were hoping to give the World Cup as a present to the people of Argentina, but that is not now possible.
Argentina is in the grip of economic crisis and the players had hoped to bring the people some relief from the traumas of everyday life.
They were the greatest team I've ever played against
But after drawing with Sweden on Wednesday, the pre-tournament favourties crashed out of the World Cup at the group stage.
Argentina's all-time record goalscorer Gabriel Batistuta has already announced his retirement from international football
And the team's exit from the World Cup will see the end of three more distinguished international careers.
And midfielder Diego Simeone has admitted that the records of himself, Claudio Caniggia and Jose Chamot will forever be tainted by the fact that they have never won a World Cup.
Simeone likened Argetina's failure this year to their capitulation in the second round as defending champions in 1982.
In Spain that year, the generation of Mario Kempes, Daniel Bertoni and Ossie Ardiles travelled to Spain and were unable to prevent holders Argentina from tumbling out in the second round.
"We looked on those guys in 1978 as role models and heroes and when they returned home in 1982 without the trophy we felt for them," Simeone said.
"Now I just feel for myself and the other three because we have given as much to the cause as anyone.
"But we will end our careers as three-time failures. It hurts, it really does," said Simeone, a veteran of Argentina's campaigns in 1994, 1998 and 2002.
Simeone says that failure to get his hands on the most famous trophy in football eight years ago in the United States is a source of bitterness.
The Argentines had played majestically, inspired by a seemingly rejuvenated Diego Maradona.
But disaster struck when Maradona tested positive for a cocktail of drugs. A brave but disheartened side went out to Romania in the last 16.
Simeone for one will nerver forget the experience.
"There we were playing beautiful football and looking like champions in waiting and blam!
"One individual's bad habits ripped the heart out of the team. I shall never forgive him," said Simeone, who has won 105 caps.
Sweden's goalkeeper Magnus Hedman was as stunned as anyone that Argentina had gone out.
"They were so skillful that it was frightening, he said after Wednesday's match.
"They are so unbelievable that you have no choice but to defend.
"In the second half we tried not to play so deep, but it proved so difficult.
"They were the greatest team I've ever played against."