Mr McBride, a special adviser in Downing Street, was removed from his job dealing with the media on a day-to-day basis in September 2008.
He had stayed on in Number 10, and was given responsibility for strategy and planning.
A Number 10 spokesman said the messages were "juvenile and inappropriate".
The spokesman added that nobody in Downing Street knew of the e-mails and that it was Mr Brown's view that there was "no place in politics for the dissemination or publication of material of this kind".
A Tory spokesman said it was absurd that advisers were "plotting smear campaigns rather than focusing on how to help people affected by the downturn".
Mr McBride said the idea to post the claims on a website had been abandoned.
In his resignation letter, he said he was "sickened" that Mr Staines had put the e-mails in the public domain and that he regretted embarrassing the government.
Cabinet Office Minister Liam Byrne said Damian McBride did the "honourable thing" by resigning
"I have already apologised for the inappropriate and juvenile content of my e-mails, and the offence they have caused," he said.
"We all know that when a backroom adviser becomes the story, their position becomes untenable, so I have willingly offered my resignation."
Mid Bedfordshire MP Ms Dorries demanded an apology from the prime minister and said she was considering legal action against Mr McBride.
"The taxpayers do not pay him to engage in this kind of behaviour," she added.
"I think the public will be very concerned that Downing Street is so desperate to hold on to power that it engages in such behaviour. "
She also questioned how Mr Brown would feel if a Conservative staff member spread untrue rumours about his wife.
Mr Grayling said Mr McBride "appears to have been setting up a machine to try and disseminate false and scurrilous rumours about opposition politicians.
"It's absolutely disgraceful. He's certainly had to go. The real question now is was he the only person involved in all of this."
But Mr Draper branded the idea of an orchestrated Downing Street campaign as "ridiculous".
He said he had been sent the comments after canvassing Labour supporters about the prospect of setting up the Red Rag blog to combat "right-wing tittle-tattle" posted on the internet.
Cabinet Office minister Liam Byrne described the material as "one private e-mail exchange between a couple of friends who were knocking backwards and forwards ideas".
"Mr McBride, having scribbled this stuff, decided that the right place for it was the waste basket," he added.
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