Response to the salam from a Non-Muslim
Question

What should we say when greeted by a non-Muslim with: As-salaamu 'alaykum?
Praise be to Allah.Can you say salam to a Non-Muslim?
It is not permissible firstly to initiate the greeting of salam to a non-Muslim . The Prophet (peace and blessings of Allah be upon him) said: “Do not initiate the greeting of salam to a Jew or a Christian. ” (Narrated by Muslim, 2167)
How to reply salam to a Non-Muslim
If one of them says As-Sam ‘alaykum meaning, may death be upon you or it is not clear whether they have said salam, then we should respond by saying Wa ‘alaykum (and upon you).
Ibn ‘Umar (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: When the Jews greet you, they say As-sam ‘alaykum (may death be upon you), so respond by saying ‘alayk (and also upon you). (Narrated by al-Bukhari, 5902; Muslim, 2461)
If a non-Muslim greets us with the correct shar’i greeting (i.e., says As-salamu ‘alaykum clearly), the scholars differed as to whether we have to return the greeting. The majority of scholars said that we do have to return the greeting, and this is the correct view.
Ibn al-Qayyim (may Allah have mercy on him) said: 
“They differed as to whether it is obligatory to return the greeting . 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 their 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 the Ahl al-Dhimmah (Jews and Christians).” (Zad al-Ma'ad, 2/425, 426)
The Muslim who is returning the greeting should respond in the manner prescribed by Shari`ah, giving a similar or better greeting, because of the general meaning of the Ayah (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-Nisa 4:86]
Ibn al-Qayyim (may Allah have mercy on him) said: 
“If the person is certain that the Dhimmi (Jew or Christian) is clearly saying as-salamu ‘alaykum to him, and he has no doubts about that, should he say wa ‘alayka as-salam or shorten it to wa ‘alayk? What is indicated by the evidence and principles of shari’ah is that he should say wa ‘alayka as-salam, because this is more just, and Allah commands us to be just and to treat others well. This does not contradict any of the hadiths on this topic at all, because the Prophet (peace and blessings of Allah be upon him) commanded us to shorten the greeting to wa ‘alayk because of the reason mentioned above, which is that they deliberately used to say as-sam ‘alaykum instead of as-salam ‘alaykum, as indicated in the hadith narrated by ‘Aishah (may Allah be pleased with her). He said, “Do you not see that I say wa ‘alaykum when they say as-sam ‘alaykum? Then he said, If the People of the Book greet you with salam, say, Wa ‘alaykum.
Allah says (interpretation of the meaning):
…And when they come to you, they greet you with a greeting wherewith Allah greets you not, and say within themselves, Why would Allah punish us not for what we say? [al-Mujadilah 58:8]
If this reason is not there, and the Jew or Christian says, Salam ‘alaykum wa rahmat-Allah, then it is only fair to respond in kind.” (Ahkam Ahl al-Dhimmah, 1/425, 426)
The hadith of ‘Aishah was narrated by al-Bukhari (5901) and Muslim (2165)
Reference:

Majmu’ Fatawa Ibn ‘Uthaymin, 2/97

For more about greetings, please see this category: Manners of Greeting with Salam
And Allah knows best.