If a non-Jew sneezes, can one say G-d Bless you for shalom purposes, as this is a common courtesy practice? Is there a better way to respond? We do not want to make a cillul Hashem if a frum Jew doesn’t respond.
It is certainly permitted to respond with “bless you” to a non-Jew, and this is of course the correct and courteous thing to do.
The question is a little disconcerting – why should this be forbidden? The Mishnah tells us that every human being is beloved, for all are created in the Divine Image, and the Shaarei Kedushah writes that we must love non-Jews as fellow human beings. Blessing a non-Jew is therefore perfectly fine, and we do so often (in prayers for the authorities and so on).
In addition the Gemara (Berachos 17a) teaches of how one must promote Shalom with Jew and non-Jew alike, and of course acting with common courtesy is a precondition.