Coetzee called the defeat in Florence "the darkest moment of my coaching career. Definitely the toughest."
He took charge this year after Heyneke Meyer opted not to seek a renewal of his four-year contract.
Meyer had faced public and media criticism for "prehistoric" kick-and-chase tactics and largely ignoring black stars in race-sensitive South Africa.
Recently appointed South African Rugby boss Mark Alexander has promised a review of a crisis-littered season that started with a first home defeat to Ireland.
"It has not been good enough," admitted Alexander, who succeeded Oregan Hoskins last month and has an in-tray full of problems, topped by the embattled national team.
Before jetting to Europe, South Africa lost to New Zealand (home and away), Ireland (home), Argentina (away) and Australia (away).
They achieved narrow home victories over Ireland (twice), Argentina and Australia.
The 57-15 mauling by the All Blacks in Durban last month was the heaviest home loss suffered by a country that has twice been crowned world champions.
Alexander refused to blame Coetzee, who made the Western Stormers the most successful South African Super Rugby side during six seasons in charge before working in Japan.
"It would be easy to lay the blame at one door or another and look for scapegoats, but it would also be an oversimplification.
"At the end of the season we will undertake a full review and (consider) what interventions may be needed to turn things around.
"It is our job to do as much as possible to make sure we are not in a similar position in 12 months' time."
But the Sunday Times criticised Coetzee, saying "he has been unable to stem the flow of pathetic results that have relegated the Springboks to a tier-two nation".
"Coetzee will point to numerous structural failings in South African rugby as the reason for the downfall, but it does not wash.
"There is disagreement on tactics among the coaching staff - the defence has been shambolic all season and the attack impotent.
"Coetzee lamented that there have been three defence coaches, but he was in charge and should have identified that area as the most important building block.
"The coach has picked players who do not appear to have the right fortitude to cope with the high pressure of Test rugby."
