Ms Collins said she had done nothing wrong and would fight to clear her name.
Her resignation was for her ministerial roles, but she will continue to be the MP for Papakura and stand in this election.
Read more of today's coverage from the Herald:
• Collins resignation: Latest updates
• Opposition parties united: Judith Collins had to go
• The email that brought down
• Full statement from Judith Collins
• Judith Collins' career in pictures
The email, from October 5, 2011, was sent to "mark", "Carrick", and one other person whose name has been redacted.
At the time Mr Feeley was under pressure for celebrating the laying of charges against failed finance company Bridgecorp with a bottle of champagne recovered from Bridgecorp's former headquarters.
Ms Collins had referred the matter to the State Services Commission, which found that Mr Feeley had had used poor judgement.
In the email, released by Mr Key, Mr Slater says that Ms Collins was "gunning for" Mr Feeley.
"I also spoke at length with the Minister responsible today (Judith Collins). She is gunning for Feeley," the email says.
"Any information that we can provide her on his background is appreciated. I have outlined for her a coming blog post about the mass staff turnover and she had added that to the review of the State Services Commissioner. She is using his review of these events to go on a trawl looking for anything else. It is my opinion that Feeley's position is untenable."
Today Mr Slater said he was simply "talking up a big game".
He said he was not lying when he used the term "gunning for".
"Embellished is a good word. It's better than a lie, isn't it? ... If Judith Collins was gunning for somebody, she'd have got them.
"I've not proved the downfall for my good mate's career. A criminal hacker mounting a political conspiracy to subvert our democracy and undermine our election has done that, and the media and all of you are complicit."
Former SFO Director Adam Feeley
Who is Adam Feeley? Read an in-depth 2010 Herald profile of the former SFO head here.
Opposition parties rounded on the resignation, saying the email confirmed the kind of dodgy tactics that were alleged in Nicky Hager's book Dirty Politics.
Labour leader David Cunliffe called for a commission of inquiry, which would have the power to require the production of evidence, to compel witnesses, and to take evidence on oath.
Ms Collins defended herself this afternoon, saying she had been subjected to a "two year smear campaign" of "false allegations and innuendos".
She staunchly denied trying to remove Mr Feeley and said she looked forward to clearing her name.
"I cannot control what other people say about me or when they leverage off my name."
She no longer wanted to be a "daily distraction" from the election campaign.
"This is an extremely distressing and disappointing time ... I will of course cooperate with any inquiry."
Mr Key said a person had passed the email to a Beehive staffer, who had given it to his chief of staff Wayne Eagleson yesterday.
He said it was a "very serious" matter, and an inquiry would investigate. More announcements on the nature of the inquiry would be made in coming days.
"The statement could be completely wrong, it could completely misrepresent the situation. But I can't let that rest unresolved," Mr Key said.
"It's not tenable for her to carry on as minister until these issues are resolved."
He said it was no secret that Ms Collins regularly talked to Mr Slater.
If National was re-elected, Ms Collins would not be a Cabinet minister, and he hinted that it would be a hard road back, even if she was cleared of any wrongdoing.
"She may need to take a little time to reflect as well ... there's been one or two slips in recent times."
Mr Feeley said he had no comment at the moment when contacted by the Herald.
Carrick Graham, a public relations specialist who received the email, also did not wish to comment. He is a central figure in Dirty Politics, which alleges he ran corporate campaigns via Whale Oil.
The email also mentions Cathy Odgers, a lawyer who used to blog as Cactus Kate. She also declined to comment.