It is not not allowed to greet someone from the other gender even through a Sh’liach & even through her husband. Lo Sasuru Acharei Ainaichem prohibits seeing anything that can generate lewd thoughts. We find more Gedorim and Siyagim (fences and safeguards) against Arayos than any other sin.
1) Even Hoezer Siman 21 S’if 7 Ein Shoialin Bishloim Isho K’lal Afilu Al Y’dei Sh’liach “V’afilu Al Y’dei Baloh” V’osur Lishloiach Lo Divrei Sh’loimim, even sending her the greeting is forbidden! 2) G’moroh Yuma 74: Omar Raish Lokish etc. & Rambam Hilchois T’shuvoh 4 Uk’var Kosvu Horishoinim, where you can see how the Yetzer has people do things that are not explicitly Osur Kdai Sheloi Y’hei Liboi Noikfoi V’yachazir Bit’shuvoh. 3) Maseches Kalloh 1 Omar R’ Elozor Kol Hshoiseh, even unintetionally. 4) G’moro Nidoh 13. Kol Hameivi….Ein Machnisin Oisoi Lim’chitzosoi Shel HKB”H & the Rambam explains V’im Poga B’machashovoh Chetoi Godoil B’harbeh 5) Chinuch 188 even if he knows he will not faulter. 6)Y’rushalmi B’rochois P’1 H’8 Omar HKB”H Im At Uhiv Liboch V’einoch Ano Yodaano D’at Dili, what A Z’chus & S’char 7) Midrash Shir Hashirim 3, 13 T’nino B’shem R’ Doiso, ….Omar Hakodoish Boruch Hu Mi Shehu Oimeid B’yetzer Shel Haznus Maaloh Ani Olov K’ilu Oimeid Bishteihem (including S’char for A”Z which is not around anymore & there is no other way to get S’char for it) 8) G’moro Makois 23: R’ Shimoin Bar Rebi….Gezel V’aroyois….Hapoiresh Meihem…..Sheyizkeh Loi U’l’doiroisov…..Ad Soif Kol Hadoirois. I can go on & on, on the Isurim,on the Oineshim, & on the S’char.