Is it obligatory for a man to greet his non-mahram female relatives when passing by them, or is it merely recommended?

If the female visitor and those with her are mahrams to this brother, then greeting them with salaam is a Sunnah. However, if they are non-mahrams (strangers), then greeting them with salaam is neither obligatory nor recommended; rather, it is permissible if there is no fear of fitnah (temptation).

Regarding the rulings of salaam between men and women:

If the woman is a wife, a bondswoman, or a mahram, then greeting her with salaam is a Sunnah, and returning the salaam is obligatory. If the woman is a non-mahram and is elderly or not desired, then greeting her with salaam is a Sunnah, and returning the salaam verbally is obligatory. If the woman is young and there is a fear of temptation by her or for her, then greeting her with salaam and her returning the salaam are makrooh (disliked) according to the Malikis, Shafi'is, and Hanbalis. The Hanafis stated that the man should return it to himself, and the woman should return it to herself. The Shafi'is explicitly stated that it is forbidden for the woman to return his salaam. A man greeting a group of women is permissible, as is a group of men greeting a single woman, if there is no fear of fitnah. Some scholars hold the view that greeting a group of women with salaam is a Sunnah if fitnah is absent. Al-Nawawi said: "If the women were a group and the man greeted them, or if the men were a large group and they greeted a single woman, it is Sunnah if there is no fear of fitnah upon him, nor upon them, nor upon her."