Referee Calum Murray sent the South Korean off in the first half of the 3-2 Clydesdale Bank Premier League defeat for hauling back Rangers midfielder Lee Wallace at the edge of the box, deeming it to be an obvious goalscoring opportunity.
A Fast Track Tribunal at Hampden this morning rejected Cha's appeal and a one-match suspension has been imposed with immediate effect which means he misses the visit of St Johnstone on Sunday.
Celtic chief executive Peter Lawwell was at Hampden on Thursday for the Professional Game Board meeting.
Responding to Cha's appeal outcome, he said: "I was at another meeting and I've just heard.
"Obviously we think it was the wrong decision and we're disappointed."
Asked if he believed the player had a fair hearing, he added: "I wasn't there so I don't know. I'm just disappointed."
It was revealed on Wednesday night the Scottish Football Association's compliance officer has written to Neil Lennon for the third time in just over a week, asking for an explanation of comments made by the Celtic manager about referees ahead of their Clydesdale Bank Premier League clash with St Johnstone at Parkhead this coming Sunday.
However, Lawwell does not believe there is a vendetta against Lennon, insisting: "Not at all."
He added: "I'm only aware of two. We are taking the appropriate advice and we will make the appropriate representations at the right time."