One woman said the cake was a "slap in the face to all the wonderful dads".
"If you want to support a "Special Persons Day", do it on any other day! So disappointed that you would make this divisive political statement," she added.
However, others said they disagreed with the backlash against "Special Person's Day", saying not all Australians had fathers in their lives.
"There may be other people in their life that may have to take that role on. I truly celebrate that you have created a cake saying "Special person's Day" because for every family that does not have a mother or father around and someone else to fill that massive role, they are 100% a very special person. Well done Woolies!," a woman wrote.
Another congratulated Woolworths on "taking into consideration other people's circumstances".
"My son doesn't have his father around a very sad day for him in fact but you turned it into a positive by providing a cake acknowledging a special person so he gave that to his uncle to say thank you for being a special part of his life, all round a happy day instead of a sad one," the woman said.
A Woolworths spokesman responded to customers over Facebook, saying the contentious photograph had been cropped out not to show other cakes that contained "Happy Father's Day" messages.
"We're currently looking into this display with our store teams. We want to reassure you that we're helping all customers across Australia celebrate Father's Day as seen from our store displays, products and recipe ideas. Cropped out of this image are a range of decorative cakes that have different messages on them, including 'Dad' and 'Happy Father's Day'. Thanks again for sharing."
- news.com.au
