Sheikh `Abd al-Rahmân b. Ibrâhîm al-`Uthmân, professor at al-Imâm University
It is obligatory to reply to the greeting of peace given by a Muslim, regardless of whether that Muslim is a man or woman.
Allah says: “If you are addressed with a greeting, then reply with a greeting that is better or return the (same) greeting.” [Sûrah al-Nisâ’: 86]
However, when replying to the greeting given by a member of the opposite sex, one must take care to reply in a suitable manner that is free from flirtation or innuendo.
The ruling is different regarding initiating salutations of peace with members of the opposite sex. It is merely permissible to do so when there is no fear of temptation. In the case where there is fear of temptation for either party, which is sometimes the case when a young man greets a young lady, it should be avoided. As long as no temptation or mischief is feared, then there is no objection to initiating greetings with members of the opposite sex.
The proof for this is what is related by Asmâ’ bint Yazîd that: “the Prophet (peace be upon him) passes us – we were a group of women – and he greeted us with peace.” [Sunan Abî Dâwûd (5294) and Sunan Ibn Mâjah (3701)]
Sahl b. Sa`d related: “There used to be an old lady among us who, on Friday, would harvest greens. When we departed from our Jumu`ah prayers, we would greet her with peace.” [Sahîh al-Bukhârî (938)]
And Allah knows best.
Source: Islam Today