Can You Initiate a Greeting to a Non-Muslim?
Question

Is it permissible for a Muslim to Initiate a greeting to a non-Muslim?
Praise be to Allah.Shaykh Ibn `Uthaymin (may Allah have mercy on him) was asked about the ruling on greeting non-Muslims . 
He replied as follows: 
“Greeting a non-Muslim first is prohibited because the Prophet (peace and blessings of Allah be upon him) said: “Do not initiate the greeting with the Jews and Christians, and if you meet them in the street push them towards the narrowest part of it.” But if they greet us, we have to respond to them , because of the general meaning of the verse in which Allah 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.” [An-Nisa’ 4:86]
The Jews used to greet the Prophet (peace and blessings of Allah be upon him) by saying, “As-sam `Alaykum ya Muhammad (Death be upon you, O Muhammad),” praying that the Messenger of Allah (peace and blessings of Allah be upon him) would die. So the Prophet (peace and blessings of Allah be upon him) said: “The Jews say `As-sam `alaykum (death be upon you),’ so if they greet you, then say, `Wa `Alaykum (and also upon you).’” 
If a non-Muslim greets a Muslim by saying “As-samu `alaykum,” then we should respond by saying “wa `alaykum (and also upon you).” 
The fact that the Prophet (peace and blessings of Allah be upon him) said “wa `alaykum” indicates that if they were saying “As-salamu `alaykum (peace be upon you),” then peace will also be upon them, i.e., whatever they say to us, we say to them. Hence some of the scholars said that if a Jew, Christian or other non-Muslim clearly says “As-salamu `alaykum,” it is permissible for us to say “ `alaykum As-salam (upon you be peace).”  
Similarly it is not permissible to initiate a greeting such as “Ahlan wa sahlan” (welcome) and the like, because that is a kind of honouring them. But if they say something like that to us, then we should say something similar to them, because the greeting should be returned in like manner and each person should be given his due. It is well known that the Muslims are higher in status before Allah, so they should not humiliate themselves in front of non-Muslims by greeting them first. 
So, in conclusion, it is not permissible for us to greet non-Muslims first, because the Prophet (peace and blessings of Allah be upon him) forbade that, and because this is a humiliation for the Muslim when he starts to honour a non-Muslim. The Muslim is higher in status before Allah, so he should not humiliate himself in this manner. But if we are greeted then we should return the greeting in similar terms. 
Similarly it is not permissible for us to greet them first with words such as “Ahlan wa sahlan” (welcome), “Marhaban” (hello) and so on, because that is a kind of honouring them, so it is like initiating the greeting of Salams with them.” (Majmu` Al-Fatawa, 3/33) 
However, if there is a need to greet a non-Muslim first, then there is no sin in that, but it should be something other than the greeting of Salam, such as saying “Ahlan wa sahlan” or “how are you”, etc. In that case the greeting is for a reason, not to honour him. (See Al-Mawsu`ah Al-Fiqhiyyah, 25/168) 
Ibn Al-Qayyim (may Allah have mercy on him) said in Zad Al-Ma`ad (2/424), concerning greeting a non-Muslim first: 
“A group of scholars said: It is permissible to greet him first if that serves a purpose, or for fear of his harm, or because of blood ties, or for a reason that requires that.”
And Allah knows best.