Di Maria joined Manchester United from Real Madrid last month for a British record fee of 59.7 million pounds (USD 97.45 million) with the Spanish club's president Florentino Perez saying the winger's demands for a new contract were too steep.
"There was a letter from Madrid that arrived at 11 in the morning of the match day, we were due to play at 4 in the afternoon... was putting ice on my leg to be in the final," Di Maria told an Argentine radio station on Friday, suggesting the club did not want him to aggravate a leg injury.
"I tore up the letter as soon as it was given to me and didn't take any notice of it, I didn't care what might happen, but it was (coach) Alejandro's (Sabella) decision," he said.
"I didn't want to make any demands of him but I also didn't want to leave the team a man short five minutes from a very important match,” he added.
Di Maria, whose goal gave Argentina a 1-0 win over Switzerland in the last 16, suffered a leg muscle injury during the 1-0 quarter-final win over Belgium and took no further part in the tournament.
Argentina lost the final 1-0 after extra time to Germany at the Maracana in Rio de Janeiro on July 13.
"I suffered a lot that morning, it was the worst thing that's happened to me in my life in footballing (terms) because reaching another final is very difficult, not to say impossible," Di Maria said.
"The letter was handed to (late Argentine FA president) Julio (Grondona) and he said if I wanted to play I should do so anyway. From what I understand it had something to do with insurance,” he added.
Later on Friday, Julio Grondona Jr accused Perez of wanting to intimidate the AFA into not fielding Di Maria, who was man of the match in Real's Champions league victory in the final against Atletico Madrid in May.
"We received the letter signed by Florentino Perez saying he did not allow Di Maria to play... We called Di Maria and together agreed to tear it up," said Grondona's son, president of Argentine first division club Arsenal founded by his father, and a member of Argentina's World Cup delegation.
"We were pressured,” he added.
Di Maria, outstanding in Argentina's 4-2 friendly win over Germany in Duesseldorf on Wednesday, denied ever wanting to leave Real and once again praised former team mate Cristiano Ronaldo for backing him to stay at the club.
"I never wanted to leave Madrid, Cristiano always supported me to stay. Selling me was very good for Madrid to recover all they had spent (in the transfer market)," he said.

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