The group has vowed to put up to 10,000 contact orders on the internet in protest at new proposals to make the family courts more transparent, which it claims do not go far enough.
Harriet Harman QC, the minister for constitutional affairs, is leading a Government proposal to open the family courts to the public and press.
But Fathers 4 Justice is arguing that her proposal to grant anonymity to the individuals involved in family cases would make a mockery of the changes.
Matt O'Connor, founder of Fathers 4 Justice, said: "If you extended this line of thinking through to its logical conclusion, then any parent who appeared in the papers should be given anonymity to 'protect' their kids.
"Anonymity and secrecy have been used as wafer-thin excuses for preventing openness for 30 years."
O'Connor, who now acts as an adviser to the organisation, has already written to Harman, calling for the provision for anonymity to be left out of the proposals.
If the group's demands are not met, it says it will press ahead with the publication of the contact orders, which it has collated from supporters over the past three years.
The identities of all those involved, including any children, would become public knowledge - in clear breach of section 97 of the Children's Act.
Jack O'Sullivan, the co-founder of Fathers Direct, said: "The systematic failures of the family courts should be exposed to greater scrutiny. But we oppose the publication of children's identities, which could be dangerous."
- Fathers 4 Justice is threatening to publish thousands of secret papers from the family courts
- The group is against proposals designed to open up the courts, which it says do not go far enough
- Harriet Harman QC, the minister behind the move, favours the retention of anonymity for anyone involved in the cases.