Islamic State territory was wiped out with the fall of Baghouz last month, but security forces have not yet been able to find Akavi or get confirmation of whether she was still alive.
The ICRC decided to name her in a public statement today as part of a public plea for any information that could lead to her location.
Prime Minister Jacinda Ardern said the Government did not want Akavi's name in the public domain, standing by the long-held position that doing so could endanger her life.
"It absolutely remains the Government's view that it would be preferable if this case were not in the public domain," Ardern told media at her post-Cabinet press conference today.
Ardern said that this position was made clear to the ICRC, but Stillhart, speaking from Geneva, said her comments came as a surprise.
"Every decision was coordinated with the New Zealand Government, with whom we have an excellent relationship, excellent cooperation, and that included the difficult decision now to go public," Stillhart said.
"We would not have made that decision without the support of the New Zealand Government."
He said he was not expecting Ardern's objections.
"I have to say when I woke up this morning and I saw that that was the information that came out of New Zealand, I was slightly surprised."
He said he met with a Government official as recently as Friday and "we were fully aligned in terms of what we are now doing, going public".
A spokesman for Ardern said New Zealand's objections were clearly made from the outset through the Foreign Minister Winston Peters' office to ICRC director-general Yves Daccord, as well as via the Ministry of Foreign Affairs and Trade post in Geneva.
He said the Government's acknowledgement of the ICRC's intentions could have been misinterpreted as support, but Stillhart's comments were inaccurate and "somewhat frustrating".
"They imply we were comfortable with going public, which we never were ... They should have been under no illusion."
Stillhart stood by the decision to go public, and did not think the apparent misunderstanding would affect efforts to find Akavi.
"I have no reason to believe that what happened today ... is going to undermine or make this quest more difficult.
"We have built a relationship of trust and confidence [with the Government] and I am absolutely sure that this will continue."
He said an attempt to rescue Akavi at the end of 2017 was very close to success.
"We believe strongly that she is actually alive ... We continue to look for her."
Stillhart said the ICRC had operational teams and networks in Syria and Iraq that were looking for Akavi, but would not go into details around what they were doing.
"Every decision, including this one, was to maximise the chances of winning Louisa's freedom."
