Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear questioner, we would like to thank you for the great confidence you place in us, and we implore Allah Almighty to help us serve His cause and render our work for His Sake.
Answering your question, Sheikh Ahmad Kutty is a senior lecturer and an Islamic Scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Muslims are to greet others as we would like them to greet us. The generic Islamic greeting is to wish for peace, which in Arabic is: as-salaamu `alaykum. We are told this is the greeting that Allah taught Adam: He was told to greet with it the angels first; he was also told, “This is the formula of greeting for you and your offspring.” (Al-Bukhari)
Having said this, there is no harm in using the standard greeting that are common to people. If you are greeting Christians you may wish them Happy or Merry Christmas. This does not in any way mean that you are adhering to the specific Christian dogmas of trinity, crucifixion, etc., unless you definitely intend it. We are not to project onto words meanings that are not commonly understood by them. Just as Christians greet Muslims happy `Eid, we can definitely greet Christians merry Christmas. Islam is all about reciprocating kindness with kindness. The Qur’anic mandate for Muslims is clear:
“But when you are greeted with a greeting [of peace], answer with an even better greeting, or [at least] with the like thereof. Verily, Allah keeps count indeed of all things.” (An-Nisaa’ 4:86)
Allah Almighty knows best.