A number of supporters are upset that the club did not elect to dismiss the Magpies boss, who was fined and suspended over the incident
The Football Association (FA) imposed a seven-match ban on the Magpies boss on Tuesday, following his altercation with the Hull City midfielder.
Subsequently, the former West Ham boss will serve a three-game stadium ban on matchdays and will not be allowed on the touchline for a further four matches.
Pardew was fined £60,000 (€72,000) by the FA and £100,000 (€120,000) by Newcastle for his behaviour.
Kevin Moore, founder of the 'United for Newcastle' supporter group, believes the punishment does not go far enough and feels Pardew should have paid for his misdemeanour with his job.
"In my personal opinion, he should have been sacked for the offence," Moore told Perform.
"I don't want to see anyone representing my football club do something like that, especially the manager of a football club, so I would have liked to see him sacked to be honest.
"It [the ban] is justifiable for what he did. At the end of the day regardless of player, fan or manager, any level of football club you just can't do what he did. Especially considering you're meant to be a role model to young kids.
"Any manager that does something like that, their reputation goes out of the window. If you did that in the street you'd be arrested, so why wasn't Alan Pardew arrested at the same time?
"It [his reputation] will be tarnished, so rightly it should be. It was a cowardly act."
Moore feels Pardew's indiscretion is one in a series of incidents to blight the club and believes Newcastle's image has been spoilt as a result.
He has called for change from top to bottom, including owner Mike Ashley, though he concedes that is unlikely to happen.
"Without a shadow of a doubt [the club's reputation is tarnished]," he added. "At the start [of Ashley's tenure] it was okay, but since then it's been laughing stock after laughing stock, now it's coming down to the manager.
"It's becoming tedious to be honest, and the club is being portrayed in a bad way. The club has a lot of history and a lot of tradition and at the moment it doesn't seem to be there. There's no soul and heart in the football club. It's disappointing.
"We've needed change [for a while]. It builds on everything they've done since taking over at the club. It's one thing after another and it's time for them all to go. Unfortunately they're here for the long run."