The Shaykh mentioned the ruling on greeting a disbeliever with other than the greeting of Islam.

The Shaykh : And there is no harm, of course, in greeting them with the like of their greeting, such as "peace" which has no significance, or "good morning" or "good evening," for example. As for saying "As-salamu alaykum" to them, this is a matter specific to the Muslim for the Muslim. How could it be otherwise? The peace, as he (peace and blessings be upon him) said in the authentic hadith, So when the Muslim says "As-salamu alaykum" to his Muslim brother, the meaning of this explicit text from the authentic hadith is, as the common people say today, "May the name of Allah be upon you," meaning that you are surrounded by Allah (Mighty and Majestic) with His care and protection. Such a supplication is only deserved by one who has believed in the Islam that brought peace. We take all of this from his (peace and blessings be upon him) saying, Then this response, as well as the act of offering the greeting, has rulings that the scholars mentioned in their explanation of this hadith, among which is: