[The ruling on returning the salām to a woman and greeting her]

The answer – and Allah is the One who grants success – is that it has come in the ḥadīth: “Women are incapacity and nakedness, so veil their incapacity with silence and their nakedness with (remaining in) their houses.” And Allah, Exalted, has commanded the woman to observe covering, and has emphasized that to the extent that He has forbidden them, Glorified be He, from even revealing the sound of their jewellery, when He said, Exalted: “And let them not stamp their feet so as to make known what they conceal of their adornment…” [al-Nūr:31].
On this basis we say: It is not permissible for a woman to initiate salām to an unrelated man, nor to return his salām if he initiates it; and likewise, it is not permissible for a man to initiate salām to an unrelated woman, nor to return her salām.
However, what we have mentioned is not absolute; for there are exceptional situations in which it is permissible for salām to be given and returned between the two sexes – namely, in what necessity and need call for. In such situations there is a condition binding upon the women, namely what Allah, Exalted, has commanded them in His saying: “Then do not be soft in speech, lest he in whose heart is a disease should covet; but speak with appropriate speech.” [al-Aḥzāb:32]

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.2