Cahill was dismissed by referee Stuart Attwell at the Emirates last Saturday for a challenge on Gunners striker Marouane Chamakh and Coyle will now be without Cahill for this weekend's trip to Aston Villa, Tuesday's Carling Cup clash with Burnley and the Premier League meeting with Manchester United next Saturday.
Coyle had support from referees' chief Mike Riley but it was not enough to persuade the appeals panel to overturn the ban.
Coyle said: "The difficulty with the appeals is the criteria laid down for the panel reviewing the red cards. Some of the text says it needs to be a clear and obvious mistake by the referee; if it's a yellow card it's not clear and obvious because it's just an interpretation.
"Maybe the criteria needs to be looked at because any football person that looked at that ... I spoke to Mike Riley, the head of referees, over the weekend who also felt it was a yellow card.
"When you've got somebody of that calibre saying that, it reinforced my belief there was a chance it would be overturned."
Coyle was already without first-choice goalkeeper Jussi Jaaskelainen, who is serving a three-match ban for his sending-off against Birmingham, and he revealed captain Kevin Davies could also miss Saturday's match at Villa Park after sitting out training this week following a clash of heads with Arsenal defender Laurent Koscielny last weekend.
Coyle said: "He hasn't trained all week and he won't train today. We'll have to monitor that one. Because it was a head knock it's of paramount importance that he's okay.
"Nobody wants to win games more than me but we'll not take a chance with any of our players' health. We'll give Kevin until the last minute but ultimately we have to make sure he's 100%."