asked because I was sitting with the Aveil and someone came in and the Aveil INTRODUCED us so I offered my hand and said “Shalom Aleichem.” The guy looked at me like I was a gnat and then looked away”
That appalls me. No one should embarrass someone in public (or otherwise). Even during Tisha B’Av,when we are not supposed to greet anyone, if someone DOES greet you unthinkingly, you are permitted to softly respond. And if the Aveil himself introduced you, you take your cues from the Aveil. I myself have slipped up and said, “Nice to meet you,” when thus introduced by an Aveil, to a relative of his who was there at the same time that I was paying a shiva call. It happens, and it is not a crime against humanity when it does.