Is it permitted to daven for a refuah shleimah for a non-Jew?
Thank you for your question.
Davening or saying tehillim for a gentile that is sick that he should get better, is permitted in many instances. If the gentile kept the 7 commandments, then one may daven for him, otherwise one may daven that he should correct himself. Another instance that we may daven for them is if the Jew will have a benefit out of the gentiles getting better, i.e. he is a business associate, customer, the gentile does him favors etc., then it is permitted. Then there wouldn’t be an issur of “Lo Tichanim”. This is especially true regarding a ger davening for his parents.
It would also be permitted if the gentile asks you to daven for him, because if he will see that you don’t it will cause hatred, and to avoid hatred it is also permitted to give things to a gentile. Besides, by davening for him it will cause a Kiddush H-shem when he gets better. If the goy did the Jew favors, he may daven for him,
Sefer Orchos Matana pg. 288-289, from the sefer Chaim B’yad ( R’ Chaim Palagi zt”l) 33, Darcei Teshuva Y:D 151-9, Leket Hakemach Hachadah 135-8. Sdei Chemed Clalim Lamed 117. Also see Yechaveh Daas 6-60, Derech Sicha Ekev pg. 544. Orchos Rabinu 1- pg. 64.