"This is an exceptionally serious matter and he needs to explain immediately what happened.
"The real question now is - was [Mr McBride] the only person involved in all of this?"
Cabinet Office minister Tom Watson was also alleged to have been copied in on the e-mails.
However, Mr Watson said he had known nothing of the "completely inappropriate" messages and had no involvement in discussions to create the Red Rag site.
"The first I was aware of the e-mail conversation that led to Damian McBride's resignation was when these were made known to Downing Street by national newspapers who had been given them by Paul Staines," he said.
Mr Byrne also dismissed suggestions of an orchestrated smear campaign.
"This was one private e-mail exchange between a couple of friends who were knocking backwards and forwards ideas," he said.
Cabinet Office Minister Liam Byrne said Damian McBride did the "honourable thing" by resigning
"Mr McBride, having scribbled this stuff, decided that the right place for it was the waste basket," he added.
Mr Staines has refused to reveal how the messages found their way to him, despite complaints from Mr Draper that they were private.
Mr McBride described the first claim as a ''solid investigative story'', but the other three as ''mainly gossipy, and intended to destabilise the Tories".
He added: ''Let's think about how to sequence these in with others'' - a suggestion that a longer-term plan to place stories was being hatched.
Claims were also made against the Tory backbencher Nadine Dorries, who says she is consulting lawyers and wants a personal apology from the Prime Minister.
Mr Draper has apologised to those mentioned in the e-mails and admitted "we should never really have considered the idea and I am sorry we did".
"All I can do is absolutely promise that these stories were just daft ideas that never - and would never have - got off the drawing board," he wrote on his blog.
He called on fellow bloggers to "commit to a new start" by avoiding personal attacks.
Alastair Campbell, Tony Blair's former spin doctor, wrote in his blog that he had been struck not just by the "unpleasantness" of the emails, but also by their "incompetence".
"McBride will be thinking that was his big mistake - writing it all down. His really big mistake was thinking it might be effective," Mr Campbell added.
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