The flat faces bred into bulldogs can cause a "lifetime of suffering" and vets are urging people not to buy one.
The bulldog has twice the health risks of other dogs, a study has found.
Urgent action is needed to reshape the breed and stop the UK joining the list of countries where the dog is banned, say Royal Veterinary College experts.
They want people to stop buying English bulldogs and two other popular breeds - the French bulldog and the pug - until breeding issues are addressed.
They're also calling on the public to stop "promoting" the dog on social media by posting and liking pictures.
The bulldog has soared in popularity over the past decade.
The breed, also known as the English or British bulldog, earned comparisons with Winston Churchill for its jowly face, and was historically seen as a symbol of courage and endurance.
A fashion for ever more extreme features, such as a flat face, wrinkled skin and squat body, has made the breed prone to health complaints, raising welfare concerns.
With their large bulging eyes and flat face, the dogs are undeniably "cute", said Dr Dan O'Neill of the Royal Veterinary College, one of the authors of the study, but their extreme body shape after years of selective breeding has become their downfall.
"For breeds such as English bulldogs where many dogs still have extreme conformations (a dog's structure and appearance) with poor innate health, the public have a huge role to play by demanding dogs with moderate and healthier conformations," he said.
"Until then, prospective owners should stop and think before buying a flat-faced dog."
The English bulldog was once a muscular and athletic breed, but over the years has become a popular pet, with a trend towards a short skull, protruding jaw, skin folds and a squat build.
The public has an important role to play in driving change by not posting pictures of the dogs on social media or liking posts, thereby "inadvertently advertising" them, Dr O'Neill said.
But he admitted the "phenomenal" popularity of the bulldog is understandable, given the psychological effect they have on us. With their large heads, big eyes and docile temperament, they remind us of babies, triggering our nurturing instincts.
"We interpret this as the dogs being cute, and this is totally understandable and, in fact, very hard to fight against as a human," he said.
"What we deem is cute from the outside, if you're living the life as that dog, is anything but cute. It is, in many cases, a lifetime of suffering."
The breeding of the bulldog is already banned in several countries, and, according to an expert working group of vets and welfare groups, including the Royal Veterinary College, the same could happen here if nothing is done.
Owners who already have one should look out for health issues such as eye problems, difficulty breathing and skin fold infections, and seek veterinary advice if concerned, they say.
Veterinary historian, Dr Alison Skipper, of King's College London, said many of the diseases linked to body shape had been known by breeders for more than a century. Responsible breeding, prioritising health, could "improve the welfare of this popular and iconic breed," she said.
And The Kennel Club said a ban risked driving the problem underground.
"We urgently want to see people choosing dogs not just because they like the way they look, which is often driven by celebrity and social media, and instead find breeders who use the health tools available and breed a non-exaggerated dog, where health comes first, as described in the breed standard," said spokesperson, Bill Lambert.
The study, published in the journal of Canine Medicine and Genetics, compared the health of thousands of English bulldogs kept as pets with that of other dog breeds. It found English bulldogs were twice as likely to have one or more disorders in a single year than other dogs.
The commonest health complaints were infections in skin folds (38 times more likely than in other dogs), an eye disorder known as cherry eye (26 times more likely), protrusion of the lower jaw (24 times more likely) and breathing problems (19 times more likely).
A recent study by the same team found pugs were also at high risk of health complaints.
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