Mr Miliband has said that the intrusion crossed a line of "common decency" and demanded an inquiry into the "culture" of the publication and its sister paper.
The Labour leader is already locked in a row with the Daily Mail over an article in which the paper claimed that Mr Miliband's father, a Marxist academic who served in the Royal Navy during the Second World War, "hated Britain".
His battle with the Mail has won support across the political divide.
Earlier, the Deputy Prime Minister Nick Clegg launched a blistering attack on the paper, accusing it of being "full of bile" about modern Britain.
The latest twist in the row centres on a memorial service for Mr Miliband's uncle Professor Harry Keen.
Labour sources said that after the service, Prof Keen's daughter was approached by a woman journalist.
Mail on Sunday editor Geordie Greig said he had not known about the plan to send a reporter to the service.
The newspaper described the move as a "terrible lapse" of judgement and said two members of staff had been suspended.