She also said Mr Key appeared unconcerned with her case.
Asked yesterday whether he would personally apologise to Ms Billingsley, Mr Key said: "I think that's been made clear by the various different authorities.
"What's far more important now is that the independent inquiry that's been established and will be run by John Whitehead actually gets to the bottom of all of the unanswered questions."
He said he did not make apologies unless there's a serious reason to.
"As I said at the time I relied on the advice that was given to me by MFAT (Ministry of Foreign Affairs and Trade)."
Asked about Ms Billingsley's claim that he appeared bored with her case, he said: "I'm just not going to engage in that discussion."
Today, Ms Logie told Radio New Zealand Mr Key was not taking the case seriously enough.
"The fact that he left the country while this was up in the air and hasn't deigned to respond to the situation proactively after he had the opportunity to hear Tania's side of the story and that now he is dismissing it as unimportant, I think is disrespectful to the people of New Zealand."
Watch: Diplomat case: McCully apologises to Key
The case was an important matter and Mr Key owed Ms Billingsley a formal apology, she said.
She also called for Minister of Foreign Affairs Murray McCully to "formally and properly" apologise to Ms Billingsley.
"Rather than just do it through the media - I think that would be a good first step."
Rape Crisis spokeswoman Andrea Black told RNZ Mr Key's lack of apology minimised the seriousness of sexual violence.
"That's a really sad reflection on one of our primary leaders in the country.
"It undermines someone who is seeking justice."
Mr McCully and MFAT head John Allen apologised to Ms Billingsley when the case became public, though Ms Billingsley said the minister's public apology was inadequate.
Rizalman was arrested on May 10 and charged with burglary and assault with intent to rape. MFAT asked Malaysia to waive immunity so he could face the charges, but behind-the-scenes communications led Malaysia to believe he could return home.
Malaysia agreed to send Rizalman back to New Zealand to face the charges, but his return has been delayed while his mental health was being assessed.
Mr Key said Malaysian Prime Minister Najib Razak spoke to him during the Prime Minister's holiday in Hawaii, and reassured him the diplomat would be sent back to New Zealand.
The independent inquiry is looking at how Rizalman was allowed to leave New Zealand, and why ministers were not informed about developments in the case until it was publicised by the Herald on Sunday.
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