Q. Is one allowed to pray for a refua shelaima or wish a refua shelaima to a non Jew?

A. Please read question 1084 hereby quoted:
“Q. Can the gabay of our shule make a mi sheberach for a Gentile (he is a good friend to one of our members) and is ill? Does it matter if he observes all seven Noahide mitzvos, or can yo recite if he is an adherent to the Catholic religion? If you can do you make a separate Mi Sheberach, since you can’t say “betoch shaar choley Yisroel?”
A. Arugas Habosem (O.H. 1: 33) addresses this question and quotes Magen Avraham (O.H. 189: 1) in regards to not including an Akum present at a meal in the horachamons of birkat hamazon., since one transgresses the prohibition of “Lo Techonem” (not yo give an Akum a free gift.) However, he qualifies that the restriction applies only when the Akum is not aware of the blessing granted. Otherwise, it is nor “bechinam” or free since he will fill obligated to acknowledge and reciprocate the honour or favour given. He advises not to include the Akum in the common Mi Sheberach recited, since the Forefathers of the Jewish nation are mentioned. Afarkasta DeAniya (O.H. 29) adds another reason from Sefer Chasidim (85) not to bless one who may have in his heart an unknown latent hatred for the Jewish people. See Tzur Ya’akov (130) in regards to blessing an Akum at a wedding. However Poskim permit in cases where avoiding the public blessing can create or cause enmity and hostility. (ibid.) Horav Shlomo Miller’s Shlit’a opinion concurs with the views mentioned, specially since the ones who usually perform such Mi Sheberachs belong to assimilationist congregations.”
Horav Shlomo Miller Shlit”a added that one may pray privately for the refua of a Gentile who keeps the Seven Mitzvos. Even if he does not, one may pray privately if he is informed of the prayer offered for him, and feels obligated and thankful, as this removes the prohibition mentioned above.

Rabbi A, Bartfeld as revised by Horav Shlomo Miller Shlit”a.