Moyes claimed the Old Trafford side were "playing referees as well as the opposition".
The Scot had described a decision not to award United a penalty as "scandalous" after an earlier defeat by Tottenham Hotspur. Moyes could now face disciplinary charges.
The Capital One Cup tie is also under scrutiny by the FA after a number of flares were set off during the match. "We will contact the clubs to seek their observations in relation to the use of flares," said a FA spokesperson.