Pirelli had hoped to revise its rears after the Spanish Grand Prix in the wake of a spate of delaminations early in the campaign.
However, its moves were blocked by Force India, Lotus and Ferrari because they feared it would affect their competitiveness.
But following a spate of blow outs at the British Grand Prix, those teams conceded that there were safety grounds for Pirelli to make changes.
Whitmarsh believes the way events panned out shows that Pirelli should have been tougher when it came to enforcing its belief that tyres needed to be improved.
"When you have made mistakes, and certainty we've made them over the years including this one, there is always a reluctance to admit them," explained Whitmarsh.
"There is always a fear of what the media will make it. But it was also clear to many that there were issues with the tyres.
"There was a reluctance to admit it - and teams will always lobby for their own competitive self-interest.
"As I said to Pirelli for quite a few weeks before Silverstone: 'You cannot listen to the teams on this one. You guys are the tyre experts; you have responsibility to give us safe tyres. You have to make the right call'.
"Pirelli got themselves a bit caught in the middle listening to the teams, especially those that didn't want anything to change."
Paul Hembery, Pirelli's motorsport director, agrees with Whitmarsh's view that his company could have pushed harder.
When asked if he felt Pirelli should have shouted more for changes when the delamination issue first came up, he said: "Probably.
"I think sometimes we have been too good citizens trying to do everything right, and every time we try and push something we get into trouble.
"We are only interested in doing our job; we are not interested in helping anyone else."
Whitmarsh thinks the fact that the grid order was relatively unchanged at Silverstone showed that teams' concern of a big fluctuation in form were unfounded – which is why Pirelli should have been allowed to make changes earlier.
"If you look at where people are now [after Germany] I don't think it has been greatly changed by this belt change, and I don't think it will be greatly changed by the construction change for Hungary.
"They will have a different shape, so there will be marginal winners and losers. It is an area that is difficult to have correlation with the wind tunnel, and that is why teams do a lot of work around the tyres at the circuit.
"I don't think anyone can accurately predict what will happen, but it will not reverse the grid or anything like that - unfortunately."