Reference: Ghaaratul-Ashritah 'alaa Ahlil-Jahli was-Safsatah v1 p115
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 [transmitted by Abu Daawood no.5204 and Ibn Maajah no.3701; graded saheeh by al-Albaanee], 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} al-Ahzaab 33:32
