Indians are influenced a lot by the western culture. They don't mind following anything which is practiced widely in western countries. So I think it is more of a cultural influence than religious.
When it comes to issues related to Muslims, Hindus find it hard to give up their prejudices against them because of a variety of reasons (partition, Pakistan, riots, Ayodhya and I'm sure there are many more) . So I don't find it surprising at all that Hindus don't send greetings to one another during Eid or Ramadan. But I guess you being an Indian already know this.
Do Muslims greet each other merry Christmas?
I'm sure many do. But Islam doesn't allow that because to say that God was born on such day is blasphemy according to the teachings of Quran.
Say, "He is Allah , [who is] One,
Allah , the Eternal Refuge
He neither begets nor is born,
Nor is there to Him any equivalent."
Quran, 112:1-4
Link to download Quran
http://www.imaanstar.com/quran.php