The show starring Leigh-Anne Pinnock and Kyro, her American Bulldog, comes as the government must ban the importation of dogs.
MP Dr Neil Hudson called the screening of the scenes “poorly judged.”
The vet added: ‘Ear trimming is a cruel, horrible and unnecessary practice which is rightly illegal in the UK.
“Celebrities and people in public life have a role to play here in not endorsing or promoting the possession of cut dogs. “
People caught cutting off their ears face a maximum prison term of six months.
However, it is legal to import dogs and breeders charge thousands of pounds for them.
Mom-to-be Leigh-Anne, 29, and her footballer boyfriend Andre Gray, 30, got Kyro from famous dog breeder Simon Davis, 34.
Last year he was given a suspended prison sentence and a 15-year ban on keeping animals after being convicted of cutting dogs’ ears with razor blades.
He was also fined £ 10,000.
Watford star Gray said the couple were not okay with Kyro getting the surgery.
Jordan Shelley of the Flop not Crop campaign tapped the BBC to cut scenes for Leigh-Anne: Race, Pop and Power.
He added: “It is vital that the media do not feed the normalization of ear cutting. “
Dr Samantha Gaines of the RSPCA said cutting the ears is painful and can lead to lifelong health and behavior problems.
A BBC spokesperson said the iPlayer show explored Leigh-Anne’s experiences of racism.
They added, “This program included shoots in his own home when his dog was present momentarily in the background.
“Shots were kept to a minimum and used only when essential to the editorial. “