Millwall supporters fought each other during their defeat against Wigan in the FA Cup semi-final on Saturday before Magpies fans clashed with police following their match against Sunderland on Sunday.
Robertson condemned the "depressing" scenes and says that they must not be allowed to spiral given the current strong state of British sport following last summer's Olympic Games.
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"Against that backdrop the last thing we want are the sort of scenes we saw on Saturday at Wembley or, indeed, up in Newcastle on Sunday. We want to absolutely make sure this doesn't snowball and become fashionable."
After struggling with hooligan elements in the 1970s and 1980s, the British game has greatly improved its reputation but Robertson claims that the weekend's events are proof that similar issues had the potential to quickly re-emerge.
"Back in the 70s and 80s we used to have the reputation that this happened time and time again," he added. "If anything, what the weekend does show is that the moment you take your eye off it, it pretty quickly reappears.
"You never entirely put the lid on all of this. But from what I've heard, it's a sort of alcohol-inspired, warm day and that's what's done it."