I have gentile friends and relatives and would like to know if there’s any issue with writing to them in the month of Elul the cordial blessing of Kesiva Vechasima Tova as we wish fellow Jews?
Yes, you may do so, and so was done by the Rebbe, so long as the Gentile is not an adversary of the Jewish people.
Explanation: Also Gentiles are judged on Rosh Hashanah and hence they too are in need of a good inscription for the year. Now, as for the question of whether it is permitted for one to wish well for a gentile and bless him for goodness, this would follow the same answer to the question of whether it is permitted to pray on behalf of a Gentile, and the answer to this question is that it is permitted to pray to G-d for blessing on behalf of a gentile, or Mumar, who does good acts for the Jewish people. This applies even if the gentile is an idolater. Thus, for example, one may pray for a gentile acquaintance or business partner who is ill, or is undergoing legal troubles. However, one is not to pray on behalf of a gentile, or Mumar, who performs evil acts towards the Jewish people. [If the gentile does not perform either good or evil for the Jewish people, it is permitted to Daven on his behalf for the purpose of Darkei Shalom or other potential benefit. The Rebbe was publicly witnessed by Dollars to bless many gentiles of all walks of life. It however requires further analysis if one may even privately pray on behalf of a gentile, when there is no benefit involved, and one is doing so simply out of his good will.] Practically, it is testified that the Rebbe once told the secretary to write this Nussach of Kesiva Vechasima Tova to a gentile, stating that also the gentiles are judged on this day. Likewise, we have a copy of an English letter written by the Rebbe in which he concludes as follows “I take this opportunity, inasmuch as Rosh Hashanah the Jewish new year is a day of divine judgment pertaining to all peoples and nations, to extend to you prayerful wishes for success in the new year.”
Sources: See regarding wishing a gentile a good year: Shulchan Menachem 3:72 footnote 1; See regarding praying on behalf of a Gentile: Sefer Chassidim 790; Chaim Bayad [Falagi] 33; Darkei Teshuvah 151:29; Yechaveh Daas 6:60; Ateres Paz 1:3; Sefer Chassidim ibid based on the fact Rebbe Yochanon in the Talmud praises and blesses Charvona; Bava Metzia 114b that Eliyahu Hanavi went to save a certain gentile from Gihenom being that he did good for the Jewish people; Chaim Bayad ibid that it is completely permitted. He brings many cases in history that Rabbanim have Davened for gentiles to recover from illness, or find favor in the eyes of the king etc. Vetzaruch Iyun as to why Davening for a gentile from whom one receives no benefit does not transgress “Lo Sechanem” [Y.D. 151:11] See Chaim Bayad ibid