I appreciate your question and your sensitivity to the issue. This question does not have an answer rooted either in law codes or standard books of etiquette. Even after a google search, my response is entirely my own.
For some your question might highlight a dilemma. Even if Jews wish their non-Jewish friends a Shanna tova, blessings for the new year, they are uncomfortable when someone greets them at Christmas or asks what Santa is bring to their house. Are these two greetings different?
I suspect (but do not know for sure) that some Jews honor their non-Jewish friends with a blessing of “shanna tova”, may you be privileged to enjoy a year of blessing and prosperity. These High Holy Days mark a time shift for the cosmos, not only for Jews – the old year with its baggage is past and the entire world has the opportunity to start anew. Why not share such high hopes for renewal? My greeting of “shanna tova” is not an attempt to convert you, merely to share the blessings of this season with you.
I am reminded of a congregant who was a patient in a Catholic hospital. Every morning, as a matter of course, the Catholic chaplain would stop by and offer to pray for her speedy recovery. She told me that she always accepted, not because she felt that his blessings were any more effective than those offered at the synagogue, but because she felt that every extra prayer on her behalf was welcome in her heart and before Heaven. It couldn't hurt. Similarly, I think there is no harm in sharing the blessing of the season with those who are close to us, even if they do not share our religious outlook.
On the other hand, such a greeting carries with it echoes of the winter holiday greetings that often cause Jewish families great concern. Well-intentioned friends, store clerks and others often ask Jewish children about their plans for Christmas, if Santa is bringing them wonderful toys, or how they will be celebrating the holidays. These questions confuse some children, anger others and present uncomfortable situations for Jewish parents. Some children are disturbed that their own holidays are invisible in contrast to the ubiquitous presence of Christmas in the public market place.
I see a distinction between these two instances of holiday greetings being shared across religious boundaries. The Christmas greetings often imply – unintentionally most of the time – the expectation that everyone does or should celebrate that holiday. Even if the context is purely secular, the notion that Santa visits every good little girl and boy, for example, places a burden on Jewish families to cope with their sense of difference.
My understanding would be that sharing New Years greetings differs in that it does not imply that everyone does or should be celebrating this holiday observance. Rather it is a reflection of a Jewish notion that the rebirth of the year offers a renewed opportunity for blessings to cover the world – whether you be Christian, Muslim, Buddhist, Hindu or Jain. It does not ask, nor expect, you to change your belief. Rather, like my congregant in the Catholic hospital, the extra blessing couldn't hurt.
Answered by: Rabbi Louis Rieser