Six drivers suffered dramatic blowouts during the race at Silverstone, with Hamilton's happening after seven laps when he was leading.
"It needs to be done straight away - it's obviously an issue," said Hamilton, who recovered to finish fourth as
team-mate Nico Rosberg won.
"It was the first time in my career I've ever felt it was dangerous."
Formula 1: Lewis Hamilton wants action on 'unacceptable' tyre failures
Pirelli motorsport boss Paul Hembery said an investigation was under way to find the cause "as soon as possible ahead of the next grand prix".
Ferrari's Felipe Massa suffered a similar blowout and Toro Rosso's Jean-Eric Vergne was also affected, his problem leading to the safety car being introduced to enable marshals to clear tyre debris from the track.
Asked whether drivers could boycott next weekend's German Grand Prix, Massa said: "I don't want to say that because I don't want to create loads of problems but this is something that for our safety we can do."
However, the suggestion of any boycott was played down by other drivers and team bosses. FIA race director Charlie Whiting said: "We need to understand what happened before we start to take action."
Hamilton, who fought back from last place to finish fourth, added: "After my incident, I was definitely nervous for the rest of the race that the tyres might go again.
"Safety is the biggest issue. It's just unacceptable really. It's only when someone gets hurt that someone will do something about it.
"It's a waste of time talking to the FIA [Formula 1's governing body] and if they don't do anything that says a lot about them."
David CoulthardBBC F1 co-commentator
"There are any number of issues which can prevent a Formula 1 car finishing a race - engine failures, gearbox failures, hydraulics, you name it. They are annoying and frustrating.
"But there are a few issues that are more serious that that. And tyre failures - like brakes failures - are right up there as being as bad as it can get.
"It is a really big worry and something urgently needs to be done about it.
"If I was still a driver, I would be very anxious about racing on those tyres again."
As well as the four high-profile blowouts, Ferrari's Fernando Alonso and Esteban Gutierrez of Sauber also had tyre problems.
Alonso, who finished third, said his right rear failed at the final corner before he made his first pit stop while Gutierrez suffered a failure of his left front tyre.
Red Bull team principal Christian Horner said: "Fernando Alonso, make no mistake about it, is a very lucky boy today to be going home. I think the FIA will get involved now because they can't afford not to."
The German Grand Prix takes place next weekend at the Nurburgring.
"We had that tyre test to develop and improve the tyre and after that test they didn't do anything about it," Hamilton, who was one of the testers, added.
When asked whether he would race on these tyres on the high-speed Spa Francorchamps track, which hosts August's Belgian Grand Prix, he replied: "I'm a racing driver so I do what I'm asked to do."
McLaren's Jenson Button agreed with Hamilton that the incidents put the drivers' lives at risk. "We've had five tyres over the last few days, it's a big issue and something that needs to be sorted out," said Button.
"[Incidents] happening at 300kph, like for Checo [Sergio Perez], is not right. It's not just dangerous for the driver in the car, it's dangerous for all the other cars.
"The cars behind [shouldn't] get hit by rubber that has metal in it. It's got to change. I don't think anything needs to be said. We all know the situation."
BBC F1 technical analyst Gary Anderson went down to Turn Four, close to where two of the blowouts occurred, and found that the ridge of the kerb was "razor sharp".
British GP highlights: Nico Rosberg wins frenetic race
Pirelli has been summoned to a meeting of the sporting working group on Wednesday to discuss the problem, but McLaren team boss Martin Whitmarsh said a solution to the problem could not wait until then.
Whitmarsh and Horner both suggested reverting to the specification of tyres used last year in time for next weekend's race.
Whitmarsh added: "F1 couldn't possibly not respond to the events of this weekend. We've been lucky no-one has been hurt."
FIA race director Whiting, who admitted he considered stopping the race, said he had not experienced such a problem before.
"I can't remember anything like this," said Whiting. "Four catastrophic failures is a first. It was quite close to being red-flagged."
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