The Shulchan Aruch in Yoreh Deah Siman 385 outlines the prohibition regarding "Sheilat Shalom" (greeting with peace) to a mourner. Does this prohibition include such terms as "Hello" "Good morning" "Greetings" and the like, or the Hebrew "מה קורה" "מה נשמע" and "מה שלומך" or is it limited to the traditional "Shalom Aleichem"?
According to Gesher haChayim 21:7:5 (I think I'm paraphrasing correctly)-
The essential issur has two components:
Normally, there is no issue if one component is missing. For example, one may still bless a person with health even with G-d's name like G-d bless you after a sneeze.
Later authorities were strict with greetings like "good morning" even without G-d's name since they essentially take the place of Shalom aleichem (a chumra not to differentiate).
There are 2 ways that I see this chumra-
One possibility is that any customary greeting, even informal ones like "what up" are substitutes for Shalom aleichem.
Another possibility is that "tzafra tav" was a universal greeting in the same way Shalom aleichem was. Nowadays there are multiple formal greetings (in English) depending on the time of day as well as very informal ones like "'suuuuuuuuuuup!" (as well as many Hebrew greetings such as "hi"). In which case, the only problem cases would be greetings with G-d's name, including just Shalom as well.