BBC director general Mark Thompson suspended the pair yesterday morning after a deluge of complaints about their behaviour, which have now reached 27,000.
Brand's decision to resign came as Ross issued an apology for his "juvenile and thoughtless remarks" which included shouting on 'Fawlty Towers' actor Andrew Sachs' answer phone that Brand had "fucked" his granddaughter, burlesque dancer Georgina Baillie.
Ross said he had not issued a statement before because he had intended to apologise "to all those offended" on his Friday night chat show on BBC One, which will not air due to Thompson's decision. His Saturday morning radio show has also been pulled from Radio 2's schedules.
Sir Michael Lyons, the head of the BBC Trust, has called an emergency meeting today to hear the results of a BBC investigation into how the calls came to be broadcast.
During the meeting Tim Davie, the BBC's director of audio and music, is due to present a preliminary report on the calls to trustees, who represent the interests of licence fee payers.
Thompson will then brief them on the latest developments, followed by a discussion on what action can be taken.
Ofcom is also conducting a separate inquiry.
Brand said in a statement that he took "complete responsibility for his actions".
"As I only do the radio show to make people laugh, I've decided that, given the subsequent coverage, I will stop doing the show.
"I got a bit caught up in the moment and forgot that, at the core of the rude comments and silly songs, were the real feelings of a beloved and brilliant comic actor and a very sweet and big-hearted young woman."
Bookie Paddy Power expects XFM to be Brand's next most likely broadcast destination with odds of 7/4, followed by the comedian's former home MTV at 5/2.
In Brand Republic's poll of 193 people 38.3% thought Brand should be sacked, 33.7% thought it was a harmless joke and 28% thought the BBC should be fined instead.