"These television commercials are simply a gentle encouragement to Australian dads, and an affirmation that they are an important figure in the lives of their children," spokesman Ben Pratt said.
"The adverts have been enthusiastically accepted and many TV stations play them all year long as a community service. They are always released in the lead up to Father's Day. Unfortunately what is a simple Father's Day message has now become a 'political' statement.
"It is extraordinary that this is where we have come to as a country; we can no longer celebrate Father's Day without being forced to look at it through the lens of the same-sex marriage debate. It's a tragedy that a political motive is now implied in any mention of fatherhood. Not everything is about same-sex marriage."
The website for the Fatherhood Foundation has been taken offline, as has the Dads4Kids Facebook page. Mr Pratt said the group had "taken the preventative step of restricting access to our website and social media channels in order to protect ourselves and our families from the expected response to our situation".
"We expect that in speaking up about this that we and those connected to us will be attacked and intimidated, and subject to the same vilification in both mainstream and social media that has been meted out to those who have stuck their head above the parapet on same-sex marriage, despite this not being the purpose of our adverts," he said.
"To be clear, it was and is not our intention to enter this debate at this time through these advertisements. And what, you might ask, is in these 'political' adverts? They feature a father singing a lullaby to his baby. It is that simple."
Despite the ad's lack of political content, gay news website Pink News has accused Dads4Kids of "years of aggressive lobbying against LGBT rights", saying it had "taken its website offline in an apparent bid to conceal itself from scrutiny".
The move sparked criticism from the 'yes' and 'no' camps, with former Australian Prime Minister Tony Abbott and Liberal MP Tim Wilson both slamming the determination.
"This Father's Day ad has been stopped from airing on TV because it's too 'political'," Liberal MP Michael Sukkar wrote on Facebook.
"It's a scary world where the role of a father can be outlawed. What next?"
Nationals MP George Christensen said the ad had "fallen victim to the suppression of free speech that goes along with changing the definition of marriage".
"So will it be too political to say the word father if same sex marriage is made legal? Will the terms husband and wife be done away with and replaced with the bland politically correct term partner?"
FreeTV has been contacted for comment.
