In the prayer introducing the bedtime shema we say we forgive everyone for everything, even in previous incarnations (gilgulim), and then say, "every Jew." I'm not sure I'm understanding it correctly, but it sounds as if all this forgiving doesn't apply to non-Jews. When it comes to the halacha on forgiveness, do they apply equally to (a Jew) forgiving a non-Jew? Or do different laws apply for forgiving non-Jews?
According to some sources, such as Chaim Vital, Shaarei Kedusha I: 5 and Eliyahu Pinchas of Vilna, Sefer HaBris II: 13, we should love all people, whether they are Jews or not (see the book Compassion for Humanity in the Jewish Tradition b R' Dovid Sears, available for free viewing on Google books, for translations of these quotes).
Specifically, R' Vital says loving non-Jews is a sign of a high spiritual level, while the Sefer HaBris tries to prove that the commandment to love your neighbor should really apply to all nations as well.
So regardless of whether the laws of forgiveness technically require Jews to forgive non-Jews in the same way, under these views it would be praiseworthy and proper to do so.
Similar teachings, such as Ramak's in Tomer Devorah that we should have compassion on all people (even all beings), would seem to compel the same conclusion.
I'm trying to track it down, but I think a Breslov chassidic rabbi from the 19th century authored a short prayer forgiving everyone no matter what, and does not make any distinction between non-Jews and Jews. I'm not sure if the prayer was intended to be used in the bedtime sh'ma.
The obligation to forgive those who have hurt you stems from the prohibition of ona'as devarim (verbal abuse) (Mishnah Brurah 606:1). It would therefore make sense that one is only obligated to forgive another Jew, since the prohibition of ona'as devarim is only to a Jew (VaYikra 25:17).
Note: See the comments to this answer.