Milijas must now serve a three-match ban after being dismissed for a challenge on Arsenal playmaker Mikel Arteta during the second half of the 1-1 draw.
The Serbia international will be unavailable for the Barclays Premier League meetings with Bolton and Chelsea as well the FA Cup tie at Birmingham.
In response to the decision, Wolves club secretary Richard Skirrow released a statement which read: "We are hugely disappointed with today's ruling.
"From the video evidence of that challenge alone, we strongly believed we had very good grounds for appeal and this was an opinion that seemed to be shared by a lot of people who have viewed the incident.
"Our view that the dismissal and three-match ban would be hugely disproportionate to the alleged offence was further reinforced by other incidents in the game.
"However, despite highlighting the inconsistency and harshness of Nenad's dismissal, we have been told that it was not deemed to be an obvious and serious error and, therefore, the original decision and resulting punishment stands."
Skirrow added: "In our view it was certainly not serious foul play.
"Prior to Tuesday, Nenad had also only six cautions in English football and no suspensions.
"Likewise, as a club, we are not serial appealers of red cards.
"In fact, on further reflection, 'disappointed' doesn't seem to do justice to the feelings of everyone at the club on hearing this decision."
Wolves manager Mick McCarthy is not set comment until his pre-match press conference at the club's training ground on Friday.
But when asked after the game if he thought Milijas' challenge deserved a sending-off, McCarthy said: "I will go and ask Stuart (Attwell, referee). He did what he saw but if he sees it again, then he will know that it can't be one (a red card).
"Nenad has got his foot on the ground, makes contact with the ball and I don't think he leaves the ground.
"Someone said to me that his studs were up. Well you generally do when you tackle with your foot, sliding in."