The paper claimed it was, in fact, owed an apology for accusations of anti-Semitism.
The intensification comes amid newspaper industry fears that the controversy could harden the UK Government's views on regulating the press in the wake of the Leveson Inquiry. The Privy Council of senior politicians meets next week to discuss what form the new tougher regulation will take.
Yesterday, Alex Brummer, the Mail's City Editor, strongly defended his paper's original article about Ralph Miliband, which questioned how far the Marxist academic's views had influenced his son under the headline "The man who hated Britain".
He said: "I don't think we need to apologise for anything. This was a piece which examined somebody's views very carefully."
Mr Brummer went on to say the tabloid itself was owed an apology after a "vicious attack" by some Labour Party figures, who had suggested its actions were motivated by anti-Semitism.
While Mr Miliband distanced himself from the claims of anti-Semitism, he stepped up his demand for the paper's owner, Lord Rothermere, to mount a full inquiry into his organisation's culture and practices after the Mail's sister title, The Mail on Sunday, sent a reporter to question his relatives at a private memorial service at a London hospital.
Lord Rothermere has since apologised to the Labour leader for that incident but has refused his demands for a wider inquiry into the way the Mail papers operate.
Mr Miliband said: "In all of this, they've never apologised for the fact they said my dad hated Britain, an idea without any foundation."
He went on: "I'm not picking a fight with the Daily Mail. I don't want to be talking about my family but I felt I had to, given what happened with my dad and what happened at my uncle's memorial service."
The Labour leader added: "What I would hope Lord Rothermere would do … is look at the wider culture and practices of the Mail and the Mail on Sunday because I don't think this is an isolated incident that has just happened to my family."
On the Dunfermline by-election campaign trail, Shadow Chancellor Ed Balls branded sending a reporter to the memorial service for Mr Miliband's uncle an "outrage" .
But Mr Brummer insisted the Mail had a rigorous culture and stressed the newspaper had a right to explore views which "were rampant in the house in which Ed Miliband was brought up, and … heard at the breakfast table every morning; it tells you why he has such a low regard for free enterprise".
Labour peer Lord Glasman, however, said there were parallels between the line the Mail was taking and McCarthyism in America, "which took any criticism of the free enterprise system as anti-Americanism".
Ahead of the Privy Council meeting, Chris Blackhurst, a senior executive at The Independent and the Evening Standard, expressed concern that the dispute would affect its deliberations.
While stressing it would be wrong for the politicians only to have one article at the forefront of their minds, he noted: "Part of the row is the press versus politicians, the Daily Mail against a leading politician, and you have to hope they put that to the back of their minds, but they are only human."
The three main political parties at Westminster, together with Hacked off, the press intrusion victims' campaign group, support tough press regulation backed by a royal charter.
The newspaper industry, including the Mail, have put forward an alternative plan, rejecting "state-sponsored regulation".