Praise be to Allaah.
If the Muslim realizes that the kaafir has said to him “al-saam ‘alaykum”, which means may death be upon you, he should respond in kind by saying “wa ‘alaykum” (and also upon you).
If he realizes beyond a doubt that he has greeted him with the greeting of Islam (al-salaamu alaykum), Ibn al-Qayyim (may Allaah have mercy on him) said: if the listener realized that the Dhimmi [Jew or Christian] has said “salaam ‘alaykum” and he is sure of that, should he say “wa ‘alayk al-salaam” or just “wa ‘alayk”?
According to the evidence and principles of sharee’ah, he should say “wa ‘alayk al-salaam”, because this is more fair, and Allaah commands us to be just and to treat others well.
(Ahkaam Ahl al-Dhimmah, 1/425, 426).
Shaykh Muhammad al-Saalih ibn ‘Uthaymeen (may Allaah preserve him) said:
These people who have come to us from the east and the west and are not Muslims, it is not permissible for us to initiate the greeting of salaam with them, because the Prophet (peace and blessings of Allaah be upon him) said: “Do not initiate the greeting of salaam with the Jews and Christians.” (Narrated by Muslim in his Saheeh).
If they greet us, then we can respond in a manner similar to that in which they greet us, because Allaah says (interpretation of the meaning):
“When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally” [al-Nisaa’ 4:86].
When they greet us with the greeting of Islam – “al-salaamu ‘alaykum” – either of the following two cases applies:
Either they pronounce the “laam” (l-sound) clearly, and say “al-salaam ‘alaykum (peace be upon you)”, so we may say “wa ‘alaykum al-salaam” or “wa ‘alaykum”
Or they do not pronounce the “laam” clearly, and so they say “al-saamu ‘alaykum (death be upon you)”, so we should say “wa ‘alaykum” only. This is because the Jews used to come to the Prophet (peace and blessings of Allaah be upon him) and greet him by saying “al-saam ‘alaykum”, without pronouncing the “laam”. “Al-saam” means “death”; i.e., they were praying against the Prophet (peace and blessings of Allaah be upon him), praying that he would die. So the Prophet (peace and blessings of Allaah be upon him) commanded us to say to them “wa ‘alaykum.”
Thus, if they say “al-saam ‘alaykum”, we should reply, “wa ‘alaykum”, which means: and the same to you, may death be upon you. This is what is indicated by the Sunnah.
But if we initiate the greeting of salaam with them, our Prophet (peace and blessings of Allaah be upon him) forbade us to do this.
(Majmoo’ Fataawaa Ibn ‘Uthaymeen, 2/97, 98).
And Allaah knows best.
For more information, see Question # 6583.
Sheikh Muhammed Salih Al-Munajjid