I recently reached a Taz in Shulchan Aruch about changing the text of birkat hamazon when a gentile is present at the table. The Taz says that blessing a gentile is Lo Techanem, and therefore one should add the words bnei bris in bentching at the part where we hope Hashem blesses everyone at the table. My question is the following: Do we follow this Taz? Is one really over Lo Tichanum for blessing a non-Jew? I have seen Rishonim talk about giving gifts to Gentiles, but never have I seen a blessing being a problem. If I live in a society where a goy won’t be offended if I walk by without saying good morning, am I over a lav for blessing the goy with good morning or have a nice a day?
It is permitted to say good morning to a non-Jew, and to bid him a nice day.
The ruling you cite is widely quoted in the name of Magen Avraham (189:1), who writes that one should not include a non-Jew in the blessing.
However, this applies specifically to including non-Jews in the blessing for Jews (Arugas Habosem 33), and not to a particular non-Jew that one knows and one wishes to bless, which is permitted (as one can derive from Shulchan Aruch, Yoreh De’ah 151:11). See also Beis She’arim (Orach Chaim 61) who writes that the idea expressed by Magen Avraham refers specifically to blessing a non-Jew among Jews, and not to blessing a non-Jew on his own (see the rationale he gives for this).
See also Sefer Chassidim writes (257), it is permitted to bless a non-Jew who is going on a journey that he should return safely (and pray for his safe return — though Sefer Chassidim cites a particular reason for this), and see also Sefer Chassidim (790).
Aside from all of the above, there is certianly no issue of behaving with common courtesy, such as bidding good morning, good day, and so on, and the Gemara praises the conduct of a Tana who would always great the other before the other could greet him–even a non-Jew (Berachos 17a).