Shulchan Aruch (EH 21:6) says that one may not ask a married woman about her welfare, even via a messenger, and even via her husband. (Based on Bava Metzia 87.) Rashi in Kidushin (70b, DH Ein Sho’alin b’Shalom Ishah Klal) writes that asking a woman about her welfare is prohibited because one thereby “makes her heart and mind familiar with him” and creates a feeling of affection within the woman which could, Chas V’Shalom, lead to sin.
Rambam Mishna Torah, Issurei Biah 21:5 says a man should not send greetings to a woman at all, not even via a messenger.