An innocent person would have no problem with a DNA test.
That isn't universally true. I could see someone being so outraged by the charge that submitting to the DNA test would be to dignify the accusation beyond what it deserves. And even if the DNA test proved paternity, it doesn't prove rape, as the entire situation could have been consensual. That's why she really needed to go to the authorities right away. She claimed it was "violent and painful" ... well, a medical exam could have backed that up, along with acquiring genetic material that could easily have convicted the perpetrator.
Don't the wrongly-accused also have rights to their good name? As far as I'm concerned, until this woman provides proof, I hold her accusations to be false, since to do so would be to take away the good name of the accused without evidence. This would not be the first time a man has been falsely accused of rape ... either due to a woman having second thoughts later, or becoming resentful after a subsequent breakup, or some other psychological issues (need for attention, etc.). Consequently, until she produces SOME independent evidence, even if this be just the testimony of someone she told about this at the time it happened ... I have to dismiss this as false. I am no saying that it's certainly false, but I have to treat it as such out of justice to the accused until such as time as she provides some evidence.