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The answer is, without a court order, No. If when your father passes away she tries to make a claim against the Estate, then through that process she would have to prove she was a child of the decedent in Probate Court.
You do not have any legal standing to Petition the court, if either your Father or alleged half-sister initiated a paternity action, it would have to be for a reason i.e. child support, etc. Since your Petition has already been denied, there is really no other recourse other than an appeal--but it is highly likely you would not prevail in this.
If your Father doubts her claim, he should make sure that he has a valid Will in place. You can disinherit children in Florida, but you cannot disinherit spouses.