Praise be to Allah.
It is not permissible for a woman to be appointed as a judge, because that is one of the positions of public authority that are restricted to men only. This has been explained previously in the answer to question no. 71338 .
But if there is no male judge, and the person has no choice but to refer the case to the court, and there is no one available to refer cases to except a woman, and if not referring the case to her would lead to loss of one’s rights or cause harm, then in that case there is nothing wrong with referring the case to her as a matter of necessity.
We put this question to our shaykh, ‘Abd ar-Rahmaan al-Barraak (may Allah preserve him), and he replied: If there is no one available except this female judge, then yes, it is permissible to refer cases to her for judgement as a matter of necessity, because Allah, may He be exalted, says (interpretation of the meaning): “So fear Allah as much as you are able” [at-Taghaabun 64:16].
And Allah knows best.