Are you allowed to daven for a refuah shleima for a non-Jew?
The Sedei Chemed (Lamed, no. 117) addresses this question, and writes that it is certainly permitted to pray for the wellbeing and good health of a non-Jew whose deeds are righteous. Even if his deeds are unknown, he writes that one can pray for his good health, adding that he should add a prayer for his deeds.
This applies to Moslems, and even to Christians, who many authorities do not consider as idolaters (see Pischei Teshuvah, Yoreh De’ah 147), and who only follow the customs of their fathers (Avodah Zarah 2).
Moreover, although the Shulchan Aruch (Yoreh De’ah 123; see also Rambam, Akum 10:1-2, and Minchas Chinuch 426) writes that one should not heal idolaters, Rav Ovadyah Yosef (Yecheveh Daas 6, no. 60) writes that this will not apply to prayer alone, which is not an actual act of healing.