Atkinson sent off both Glenn Whelan and Marc Wilson, as well as Stoke boss Mark Hughes, in the space of five first-half minutes as Stoke surrendered an early 1-0 lead to suffer a heavy defeat.
Stoke's complaints extend beyond the dismissals as there was an apparent handball in the build-up to Loic Remy's equaliser while Hatem Ben Arfa appeared to carry the ball out of play before pulling it back for Yoan Gouffran to score.
While Hughes is facing an FA charge of improper conduct, Coates is planning to make his own grievances known.
"The performance of the referee and his officials was a disgrace, and we will certainly be complaining to the relevant authorities," he said in the Daily Telegraph.
"We cannot accept what happened to us at Newcastle without making some form of representation. It's then up to the officials what they do, but we found it hugely disappointing because there were so many poor decisions.
"There was no common sense applied and I don't think either of the yellow cards for Glenn Whelan were deserved. For one of the goals I could clearly see the ball was out of play, sitting high up in the stands.
"The sendings-off completely changed the game. We were playing ever so well up to the dismissals, leading 1-0, but when you're down to nine men you've got no chance. It's not a case of sour grapes, we simply cannot accept what happened."
Saturday December 28
12:45 West Ham United v West Brom
15:00 Aston Villa v Swansea City
15:00 Hull City v Fulham
15:00 Manchester City v Crystal Palace
15:00 Norwich City v Man Utd
17:30 Cardiff City v Sunderland
Sunday December 29
13:30 Everton v Southampton
13:30 Newcastle United v Arsenal
16:00 Chelsea v Liverpool
16:00 Tottenham v Stoke City