The Prime Minister described the pair's behaviour on Brand's BBC Radio 2 show as "offensive and inappropriate" in what could be seen as a direct challenge to the BBC to discipline or suspend the pair.
Ofcom has also launched an inquiry, which could lead to a fine and compulsory on-air apology. The regulator has previously fined Kiss FM £175,000 for a series of transgresssions including a prank call broadcast without the victim's consent.
Brown said: "This is clearly inappropriate and unacceptable behaviour, as is now widely recognised. Ofcom have said they will investigate the matter and it is for the BBC, the BBC Trust and Ofcom to take any appropriate action."
Opposition leader David Cameron also joined in the row by questioning why the recorded show was ever broadcast.
Brand and Ross left a series of messages on 'Fawlty Towers' actor Andrew Sachs' answer phone during a pre-recorded show claiming Brand had slept with his granddaughter, 23-year-old burlesque performer Georgina Baillie.
They also went on to joke that the shock of the revelation might encourage the actor to kill himself.
The calls were broadcast two days later on Brand's Saturday night BBC Radio 2 show on October 18 after senior producers had approved them without Sachs' approval.
The BBC released a statement saying that their behaviour was "unacceptable and offensive" and the two entertainers have also apologised.
The corporation is carrying out an internal inquiry headed by Tim Davie, director of audio and music, which will be deliverd to the BBC Trust's editorial standards committee. Davie yesterday became the first BBC figure to speak about the incident.
The Sun has published an interview with Sachs's granddaughter today in which she says the family is considering a formal complaint to police.
Baillie, a burlesque performer with dance troupe the Satanic Sluts, who also tried out for The Sun's Page 3, admitted that she slept with Brand three times after they were introduced by a mutual friend in 2006.
She called for the pair to be fired and said: "What's funny about humiliating a lovely old man who has never harmed anyone in his life?"
Ross is on an £18m three year contract, the highest in the history of the BBC, while Brand receives £200,000 a year for his weekly radio show.