Poulain also challenged the DHB she sits on over the incident, claiming the DHB had "failed significantly in its duty of care to provide safe and adequate care to mother and child in this situation".
Her criticism did not go down well with Atkinson, who said she "breached the board code of conduct by publicly criticising the DHB and the staff that work".
Since the incident, Poulain has missed three board meetings in a row - the latest being on Wednesday.
Poulain defended her attendance record and said it had been "near perfect for the last six years before this incident".
Atkinson received the text message from Poulain at 9.59am - prior to the scheduled 10am start of the meeting advising her intention to formally resign.
In the text message, she said: "Morning. My apologies for today's meeting. I was intending to come however I have been considering my options and have decided to resign, effective immediately. I will issue [an] official statement to you/board as soon as possible. Regards, Jacoby."
Atkinson said the normal process of resignation would be a letter to the Minister of Health and copied to the DHB.
He said Poulain has "never been shackled and the appropriate place to raise the issues is around the board table, not in the media".
"It's quite sad really. She's in a unique position in governance, as a young Māori lady, with a lot of intelligence and ability.
"It's just so sad that she's chosen not to follow the right pathway to resolving her concerns."
Atkinson said he had arranged two "special" board meetings prior to the May and June meetings with the full board and Jacoby to "discuss and resolve the issues and move forward".
However, at both, he received similar type texts as Wednesday's, moments before they were due to meet, to say that she had decided not to attend.
"The board were prepared to move forward, put the issues behind us, and were looking forward to her attending the meeting [on Wednesday] and were somewhat shocked really to get that sort of text," Atkinson said.
He said he had spoken to the ministry with regard to the options available to him as chairman, however no steps had been taken to remove Poulain as a board member.
Atkinson said he "never had any concerns around her right to talk about issues that she's concerned with". Instead, the concern to the board was to operate within both the state services and the board code of conduct.
These require people in a Crown entity "to behave in a way that does not criticise the organisation that they are a member of", he said.
Atkinson believed it was time to move forward.
"We've operated without her for the last three months. We've got some significant challenges ahead of us in the health status of our community and the board will continue to work positively through to its last meeting in November."
On Thursday evening, Atkinson had not received a formal letter of resignation.
