A High Court judge had already ruled that Ben Butler was to blame for his daughter's death, it can now be reported.
Mrs Justice Eleanor King analysed issues relating to six-year-old Ellie's death at a private hearing in the Family Division of the High Court in London in June 2014.
Bosses at Sutton Council social services had asked Mrs Justice King to make "findings of fact" to help staff take decisions about the future of a younger sibling, following Ellie's death and Butler's arrest.
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Mrs Justice King concluded that, on the balance of probabilities, Butler had been "responsible for Ellie's death".
The judge said Ellie had suffered a skull fracture.
She explained in a ruling: "I am satisfied so that I am sure that Ellie died as a result of [her] father either hitting her on the back of the head with the leg of [a] child's table, or swinging her with such violence that her head came so forcefully into contact with the table leg that the leg broke and she sustained the skull fracture from which she died."
The judge said her decision could not be reported while Butler's trial was taking place in case jurors were influenced.
Her full ruling has yet to published.
Butler's lawyer, Di Middleton, said he "disputes the accuracy" of Mrs Justice King's finding.
She told the Old Bailey it did not reflect all the evidence that had been heard in the criminal trial.
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