The Argentinian winger completed his £59.7million transfer to Old Trafford on Tuesday when he put to pen to paper on a five-year deal.
United smashed the British transfer record to sign Di Maria from the European champions with a fee that eclipses the previous British transfer record of £50million, which Chelsea paid for Fernando Torres in 2011 and it also exceeds the £37.1million sum United paid for Juan Mata in January.
Di Maria has sent an open letter to Spanish sports newspaper Marca in which he addresses the issue of his move to England.
Writing to Real's fans, Di Maria said: "Unfortunately today (Tuesday) I have to go, but I want to make clear that this was never my desire.
"As anyone who works, I've always wanted to progress.
"After winning "La Decima" (Real's 10th European Cup), I went to the World Cup with the hope of receiving a new contract, which never came.
"Many things and many lies were spoken.
"They always wanted to attribute the initiative to me to leave the club, but that was not the case.
"The only thing I asked for is something that would be fair.
"There are many things that I value and many of them have nothing to do with my salary.
"I hope to progress at Manchester United, one of the biggest clubs in the world, where I wish to make history."
Di Maria told Real supporters that he will treasure his time at the club.
The 26-year-old, who moved to Madrid in 2010 from Benfica, said: "It is impossible to capture all I have lived through in a few lines, but I want this letter to communicate what I feel at the moment of parting.
"In these four years I had the honour to wear this shirt I feel nothing but pride for what I went through and achieved along with my team-mates.
"I want to apologize for the times I was wrong, on or off the pitch.
"I sweated for this shirt in every game as if it was my last. There were times when things went for me and others not, but what I can say is that I always gave everything.
"I take forever in my memory that time in the Supercopa when I jumped at the Bernabeu (stadium) and I received from the fans immense affection.
"Thank you. I still have goose bumps.