David Cameron has welcomed the resignation of Rebekah Brooks but made clear that the former News International chief executive should still give evidence to the House of Commons culture select committee next week.

The prime minister, who was a guest of Brooks at her Oxfordshire home days after stripping Vince Cable of his responsibilities for the BSkyB bid in December, was told of her resignation at Chequers shortly before a planned meeting of the cabinet.

His spokesman said: "He thinks it's the right decision. He said the other day he would have accepted her resignation."

Asked whether Brooks should still give evidence to the culture select committee, the spokesman said: "I don't think he would change that. I presume the committee will still want to see her."

Ed Miliband also welcomed the resignation but said that Rupert Murdoch still had questions to answer. The Labour leader, who originally called for Brooks to resign on 5 July, said: "It is right that Rebekah Brooks has finally taken responsibility for the terrible events that happened on her watch, like the hacking of Milly Dowler's phone. No one in this country should exercise power without responsibility.

"But as I said when I called for her resignation 10 days ago, this is not just about one individual but about the culture of an organisation. Rupert Murdoch says that NewsCorp has handled these allegations 'extremely well'. He still hasn't apologised to the innocent victims of hacking. He clearly still doesn't get it. When he comes to House of Commons next week, people will expect him to start taking some responsibility and apologise for the illegal actions which happened in his organisation."

A spokesman for Nick Clegg said: "This is the right thing for Rebekah Brooks to have done. It is an important first step in cleaning up this mess. People will, rightly, expect Ms Brooks to come to the select committee next week to give evidence.

"People still need answers. She owes it to the victims of phone hacking and the country at large to explain her role in what happened."

The prime minister's spokesman declined to say whether he regrets having met Brooks on a regular social basis. They are members of what has been dubbed by political commentator Peter Oborne as the "Chipping Norton set".

"Look, he's been very clear about the issues relating to the relationships between politicians and the media and the fact that in the past that has been too cosy. That is something the inquiry will look at."

The spokesman was asked whether the prime minister regarded Brooks as a friend. Cameron said last week that he regarded Andy Coulson, his former communications director who was deputy to Brooks at the News of the World, as a friend.

"I'm not going to get into a long conversation about this," the spokesman said. "He has spoken about his relationship with Andy Coulson a lot in the past few days. He has spoken about the relations between politicians and the media. He has made his position very clear."

• If you wish to comment on the phone-hacking row, please visit our live blog here