What is the ruling on a wife going out onto the balcony with her head uncovered, justifying it by the distance between houses and the fact that no one can see her, despite her husband's warning about the possibility of passersby seeing her?

You must guide your wife and prevent her from committing sins against Allah, because that is your responsibility. Allah says: "O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones." (Quran 66:6). And the Prophet, peace be upon him, said: "All of you are shepherds, and all of you are responsible for your flock... And a man is a shepherd over his family and is responsible for his flock." She must obey you in this, as your obedience is obligatory, and she is originally commanded to observe the hijab (definition) if she is in a place where a non-mahram (definition) man can see her.