Di Maria joined Manchester United from Real last month for a British record fee of 59.7 million pounds 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 Argentine radio station America 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."
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.