Reference: Ghaaratul-Ashritah ‘alaa Ahlil-Jahli was-Safsatah – Volume 1, Page 115
Question: Is it permissible to extend the greeting [of as-salaam ‘alaykum] to a Muslim woman without looking at her and talking to her [beyond saying as-salaam ‘alaykum]?
Response: The Prophet ﷺ used to extend the greeting (as-salaam ‘alaykum) to the women and quite possibly raise his hand as a [greeting] sign, so there is no harm in saying “as-salaam ‘alaykum” and she [responds by] saying “wa ‘alaykum as-salaam” – no harm in this. And Allaah سبحانه و تعالى says regarding the women:
{…but speak in an honourable manner}, soorah al-Ahzaab, aayah 32
