Michael and Kellie Bloomfield, from Croydon, took their beloved tanned, six-year-old dog, Kitch, to Vets 4 Pets, in Lower Addiscombe Road, on Tuesday for the nail trimming.
In the past, he had worn a muzzle as he could be anxious but this time the family say Kitch was given a muzzle too small for his face meaning his mouth could not open.
The couple claim the vet's assistant put her full weight onto the six-stone pooch to prevent him from moving as he was straining to get up.
Mother-of-three Kellie, 39, said: "They got about four or five muzzles out that didn't fit him but they decided to put this long, tube one on him, the kind you'd put on a greyhound.
"The vet cut another nail and that started bleeding. By the time he had cleared the blood up, I wanted him to stop and he took the muzzle off him.
"My dog's tongue was blue and I said to them 'you've killed my dog'. I'm a lifeguard and I know that means he had a lack of oxygen. He needs to breathe through his mouth.
"It's diabolical and I am devastated. He was my baby and part of our family. I was totally numb."
The assistant took the dog out of the room and when she returned she told the couple Kitch had died of a heart attack.
Michael, 57, then came to meet Kellie and was also told the same by the vet. The couple have asked for an independent autopsy with another vet on Kitch's body and they expect to hear results in the next month.
Michael, a HGV driver, said: "It's unbelievable. Kellie was hysterical. She witnessed it all.
"I spoke to a member of staff at the vets' and she's come down on the side of the vet and the assistant and said there was no excess force used. If he was struggling you shouldn't do it.
"If you're taking your dog for an overgrown toe nail you do not expect the wrong muzzle to be used and (your dog) being laid on top off to have only one nail cut. Just, one nail. It doesn't make sense what they have done.
"I said if you went for a manicure would you expect to come out dead. It was just one clip."
Kellie, also a special needs nurse, said: "It should've only taken a minute. But I was in there for about five minutes but it felt like forever.
"The assistant said he had weed on him but it must have been the fluid coming out of him as he had died. He was stone cold dead."
The couple named their dog after Millwall legend Barry Kitchener. The dog was attacked several years ago by two other dogs, leaving him anxious in stressful situations.
The family, who have three children, made headlines last year when football-mad Michael named their newborn son "Bermondsey Millwall Den" after his favourite club.
A spokeswoman from Vets 4 Pets said: "We were upset to learn of the sudden death of Kitchener during a visit to the vet on Tuesday and offer our sincere condolences to his family.
"We have promised a detailed investigation including a postmortem examination so that we can provide a full explanation to Kitchener's family."