Westgarth, 31, won his fight against Dec Spelman in Doncaster on Saturday but looked uncomfortable as he was interviewed ringside after his points victory.
He was taken to Royal Hallamshire Hospital but his condition deteriorated shortly before arrival and his death was confirmed by his promoter Stefy Bull on Monday.
It was the first death in professional boxing since Canada's Tim Hague died two days after a fight last June.
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A year earlier Scotland's Mike Towell died from a brain injury following a bout.
Smith, a former boxer and now the British Boxing Board of Control's general secretary, reacted with shock to the latest death in a sport often criticised for its brutal nature.
"It's terrible for the sport and terrible for the family and we send our condolences to Scott's family," Smith told Reuters.
Smith said the sport had made huge advances in safety since the 1980s and that Britain had an enviable record.
"We are one of the most forward-thinking commissions in the world regarding medical aspects," he said. "Some people don't like us because they say we are too strict.
"This is a tough tough sport and we try to make it as safe as possible but you can't make it 100 percent safe," he added.
"But that doesn't take away from the fact that this is a disaster, but more importantly it's a disaster for his family and that's the most important thing."
Events such as Westgarth's death inevitably lead to soul-searching by those in the sport and fuel the debate about the sport's ethics.
"We all do this for a reason. But that doesn't take away the feeling I have today of 'why am I doing this?'," Smith said.
"We will continue. If it's not me it will be someone else. But it makes you question why you are bothering."
(Reporting by Martyn Herman, editing by Ed Osmond)
