He told the Herald today that he had not received any help, was struggling to cope and had felt suicidal.
Falconer said he had left Christchurch while his son was still in the hospital, a decision he said was "stupid" but was a reaction to the stressful situation.
"Every day I think about my son and what has happened it makes me very sick every day. Did I take on to much," he said in an email.
"Who knows, have there been mistakes made while handling this situation - yes "
The baby boy, who was born by emergency caesarean at 33 weeks after his mother's heart stopped beating, is now being looked after by a family member of his late mother, Joanne Manera, Stuff reports.
Manera is also looking after the child's two half siblings, who are not Falconer's biological children.
Falconer earlier told Newstalk ZB his partner planned to be a stay-at-home mum, and didn't have a job in the 12 months leading up to the expected date of delivery so she wasn't entitled to paid parental leave, according to the Ministry for Business, Employment and Innovation.
He had been supporting his family, including his late partner's two other children, by working but that was no longer a sustainable option after her death.
His employer and colleagues had been very supportive, he said.
Lees-Galloway said the case had exposed a gap in the legislation.
"More immediately I'm very pleased to say that we have found a solution for this particular gentleman and his family.
"[The Ministry of Business, Innovation and Employment] will make an ex gratia payment to him to the equivalent amount that he would have received under paid parental leave."
Lees-Galloway said this was the first time this had happened.
Prime Minister Jacinda Ardern said she was so concerned about the case that she personally read through the legislation that morning.
"I think everyone wants to see that support in place and I have to say I was baffled when I read it," she told reporters at Parliament.
"No one would agree that this is a good outcome."
WHERE TO GET HELP:
If you are worried about your or someone else's mental health, the best place to get help is your GP or local mental health provider. However, if you or someone else is in danger or endangering others, call police immediately on 111.
OR IF YOU NEED TO TALK TO SOMEONE ELSE:
• LIFELINE: 0800 543 354 or 09 5222 999 within Auckland (available 24/7)
• SUICIDE CRISIS HELPLINE: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• YOUTHLINE: 0800 376 633 ,free text 234 or email talk@youthline.co.nz or online chat.
• NEED TO TALK? Free call or text 1737 (available 24/7)
• KIDSLINE: 0800 543 754 (available 24/7)
• WHATSUP: 0800 942 8787 (1pm to 11pm)
• DEPRESSION HELPLINE: 0800 111 757
• SAMARITANS – 0800 726 666.