Swedish comedian Olaf Falafel won Dave's prestigious award for "Funniest Joke of The Fringe" award with this vegetable-based pun on Monday: "I keep randomly shouting out 'Broccoli' and 'Cauliflower' - I think I might have florets."
But Tourette's Action said it was "so disappointed" by the gag winning the accolade.
The chief executive of the charity Suzanne Dobson said the "rubbish" joke had brought "shame on Dave", reports the BBC.
Ms Dobson said the charity had been about to launch a campaign to encourage the public to stop using Tourette's as a punchline.
She said: "Humour is a great way of educating people - but not only is it not funny to poke fun at people with Tourette's, it's not even that funny a joke, is it?"
Ms Dobson urged Dave and the Swede to apologise, adding: "I would ask them to walk in the shoes of somebody with Tourette's for a day, and then come back and tell me if they find it quite so amusing."
Falafel, who is also a children’s book author, is performing at the Fringe for the seventh year in his show One Giant Leek For Mankind.
He said: “[The Fringe] like pregnancy – you ?go through a lot of pain and then the child is born and enough time passes and you forget about the pain and decide that you fancy having another one, but straight after you’re like, ‘I’m never doing that again’.”
On winning the award the comic - who claims to be "Sweden's 8th funniest" comedian - said: "This is a fantastic honour but it's like I've always said, jokes about white sugar are rare, jokes about brown sugar... demerara."Earlier, Inbetweeners star and comedian James Buckley took a swipe at the Fringe's best joke.
In a tweet Buckley wrote: "I’ve just read the joke that won funniest joke award at the Edinburgh fringe this year. We really do live in a time where you aren’t allowed to joke about anything."