Stoke were incensed last month when Wenger accused Shawcross of kicking Tottenham goalkeeper Heurelho Gomes at the Britannia Stadium at the end of August.
The Arsenal boss has since said Stoke should not take his comments personally but Coates is angry the Frenchman has not apologised for his claim against Shawcross.
He told the Stoke Sentinel: "A statement has been made that is factually incorrect. All the manager has to say is he is sorry, he has made a mistake.
"I realise he is a great manager, but we can't let managers, however eminent, say things which are not true that could damage Ryan's career.
"For that reason, we are asking for this to be rectified.
"We have made a complaint to the Premier League, so it is in their hands."
Coates also believes Stoke are unfairly stereotyped as a dirty team by the likes of Wenger and was amused when told of this season's Opta statistics.
Stoke have committed slightly more fouls than Arsenal (78 to 74) in the Premier League this season and have 12 bookings to Arsenal's 11.
But Stoke have yet to pick up a red card, unlike the Gunners who have had Laurent Koscielny and Alex Song sent off. Dean Whitehead has committed the most fouls for Stoke, with 12 in six games, compared to Song's 13 in five games for the Gunners.
Coates said: " It is an interesting comparison with Arsenal's and confirms our view.
"Arsene Wenger is always complaining and trying to influence officials, but the facts don't bear him out.
"We have a reputation for being a strong, physical team, but the media stereotypes teams."
And Coates believes that stereotype is costing the team on the pitch.
He added: "We don't get the rub of the green. We had a penalty given against us at Newcastle on Sunday - and I'm not saying it wasn't a penalty. But we should have had one at the other end, only Matthew Etherington stayed on his feet rather than fall over, so nothing happened. But that's the way we try to play."
Source: Team Talk
Source: Team Talk