As far as greeting the People of the Book, Jews and Christians, is concerned, Sheikh M. S. Al-Munajjid, a prominent Saudi Islamic lecturer and author, states the following: “Basically, it is not permissible to initiate the greeting of Salam to a non-Muslim. The Prophet, peace and blessings be upon him, said: “Do not initiate the greeting of Salam to a Jew or a Christian…” (Narrated by Muslim)
If one of them says “As-Samu `Alaykum” – meaning, may death be upon you – or it is not clear whether they have said “As-Salam” or “As-Sam”, then we should respond by saying “Wa `Alaykum” (and upon you).
On this issue, Ibn `Umar, may Allah be pleased with them both, quotes Allah’s Messenger, peace and blessings be upon him, as saying: “When the Jews greet you, they say ‘As-Samu `Alaykum (may death be upon you),’ so respond by saying ‘`Wa Alaykum (and also upon you).’” (Narrated by Al-Bukhari and Muslim)
However, if a non-Muslim greets a Muslim with with the correct Islamic form of greeting (i.e., by saying ‘As-Salamu `Alaykum’ clearly), the scholars differed as to whether the Muslim has to return the greeting. The majority of scholars maintain that Muslims do have to return such greeting, and this is the correct view.
This is what is clear in the statements of Ibn Al-Qayyim, may Allah rest his soul in peace: They (Muslim scholars) differed as to whether it is obligatory to return the greeting or not. The majority said that it is obligatory and this is the correct view. A group of scholars said that it is not obligatory to return such greeting, just as it is not obligatory to return the greeting of those who follow Bid`ah. But the correct view is the first one. The difference is that we are commanded to forsake the followers of Bid`ah by way of rebuke and to warn others about them, which is not the case with Ahl Adh-Dhimmah (Jews and Christians). (Zad Al-Ma`ad, 2/425, 426)
In returning the greeting, the Muslim should respond in the manner prescribed by Shari`ah, by giving a similar or better response, because of the general meaning of the following Qur’anic verse:
“When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally.” (An-Nisa’: 86)
Ibn Al-Qayyim, may Allah rest his soul in peace, said: If a Muslim is certain that the words used by Dhimmi in greeting him are ‘As-Salamu `Alaykum’, should he say ‘Wa `Alayka As-Salam’ or shorten it to ‘Wa `Alayk’? According to the principles of Shari`ah, he should say ‘Wa `Alayka As-Salam’, because this is more just, and Allah commands us to be just and to treat others well…” (Ahkam Ahl Adh-Dhimmah, 1/425, 426)”