Poyet's side, already 1-0 down at the time, were reduced to 10 men in the 36th minute of the Barclays Premier League contest when referee Kevin Friend deemed a sliding tackle from Wes Brown on Charlie Adam to be worthy of a straight red card.
Brown, who was travelling at considerable speed, won the ball with the challenge and replays suggested any contact between the two players had been minimal.
Friend's call drew an angry reaction from Poyet and, when asked about the incident after the match, the Uruguayan manager said: "I would like the referees' association (the Professional Game Match Officials Board) to apologise to me this time.
"They did it last week. I think it is fair - they called a British manager and I think it is time for them to call a foreign manager.
"Then, the referee will have said it was not a red card and I don't need to appeal."
Riley, who heads the PGMOB, recently contacted West Brom boss Steve Clarke to apologise for referee Andre Marriner's contentious decision to award a late penalty to Chelsea in Albion's league game at Stamford Bridge on November 9, a spot-kick which was converted by Eden Hazard and saw the contest end 2-2.