Protesters should show respect to Lady Thatcher on the day of her funeral, David Cameron has said, as he defended the scale of the event as "absolutely fitting and right" for a prime minister who achieved "extraordinary things".

In an interview with BBC Radio 4's Today programme shortly before travelling to St Paul's Cathedral, Cameron sought to illustrate the momentous impact of Thatcher by saying: "We are all Thatcherites now."

But Cameron indicated he was looking to the future by sidestepping questions about whether he is personally a Thatcherite. He also distanced himself from the more divisive side of her legacy by saying he was seeking to give a "big boost" to social renewal.

As protesters planned to register their opposition to Thatcher on the funeral route, Cameron said people had a right to disagree. But he said they should appreciate that the nation was paying tribute to the longest-serving prime minister of the 20th century.

Cameron said: "Of course people have a right to disagree and take a different view. But when you are mourning the passing of an 87-year-old woman who was the first woman prime minister, who served for longer in the job than anyone for 150 years, I think it is appropriate to show respect. There will be times, there have been times, when political figures who have died that I might profoundly disagree with. But when they pass you show respect and do so in a dignified way."

He defended the costs and scale of the funeral after the Today programme presenter Evan Davis said some people feared there had been a "tampering with the constitution" to give Thatcher "something akin to a royal funeral".

Cameron said: "She achieved some extraordinary things in her life. I think what is happening today is absolutely fitting and right. Looking from overseas I think people who respected and revered Margaret Thatcher and what she did – I think people would think we had taken an extraordinary view if we somehow didn't commemorate this."

Cameron agreed the funeral would not be a political or triumphalist event, saying: "It will be quite a sombre event. But it is a fitting tribute to a great prime minister respected around the world and I think other countries around the world would think Britain had got it completely wrong if we didn't mark this in a proper way."

The prime minister said Thatcher's political settlement had been accepted by all the mainstream parties when he declared that everyone was a Thatcherite. But he was careful to avoid saying he is personally a Thatcherite.

Cameron said: "In a way, we are all Thatcherites now. One of the things about her legacy is that some of those big arguments that she had everyone now accepts. Nobody wants to go back to trade unions that are undemocratic or one-sided nuclear disarmament or having private sector businesses in the public sector."

But the prime minister made clear he would not repeat Thatcher's mistakes as he said he would place greater emphasis on social renewal. "I have always felt it is important you learn from all of political history. What we have needed to do is take that great inheritance and then add to it. As well as an economic renewal there has been a need for a great social renewal. That side of Conservatism needs to have a big boost and that is what I have tried to do over the last seven years."

Cameron's careful remarks show that he understands that her legacy is a mixed blessing for the Tories. He drew a distinction with Thatcher by placing himself in the mould of Lord Heseltine, whose leadership challenged ended her premiership, in devolving powers to Britain's "great cities".

"It is very understandable why Margaret Thatcher and her governments had to take quite such a tough approach to local government. You had these loony left authorities running Lambeth [in London], Liverpool and elsewhere. So it wasn't really an age for decentralisation. But I have always felt that a true Conservative should be following the Thatcher logic of giving people more power over their lives, you should be giving our great cities and our local authorities greater power and greater control."

Cameron said the best approach to Thatcher's legacy for Conservatives was voiced by the chancellor, George Osborne, who said everyone lives in the shadow of the late prime minister and the party should embrace that.

The prime minister said: "She was an extraordinary leader. She still casts a great influence on the Conservative party today. But what is important is that we remember all of Margaret Thatcher and all the way she approached politics."