This question is different than others I've read regarding mourning a Non-Jewish parent. I'm specifically interested in the case of a Noahide father and a Jewish mother who have divorced. What I mean by Noahide is someone who professes belief in HaShem, but does not attend any non-Jewish religious services or engage in any conduct forbidden to a non-Jew by the Torah. Going beyond this, the Jewish child was raised Jewish by the non-Jewish father, and through the father's active support eventual became religiously observant. What would be the proper way to mourn such a parent?
You are asking whether a Jewish son should mourn his righteous non-Jewish father to whom he owes gratefulness for having raised him and supported his Jewish growth.
It is clear that there is no halakhic obligation to mourn for a non-Jew, at the same time certain mourning rites might be appropriate.
R David Brofsky in his book Hilkhot Avelut (p. 109) writes explicitly
The Torah does not recognize the relationship between a person born of a Jewish mother and his non-Jewish father
At the same time, for converts (and it also applies in this context), he continues
of course, a convert should treat his or her non-Jewish parents with appropriate respect (SA YD 241:9) [...] as an expression go gratitude for their having brought him into this world. (Igrot Moshe YD 2:130)
He then goes on to discuss which practices one can voluntarily adopt in such a case and suggests reciting Kaddish (R Ovadia Yosef in Yehave Daat 6:60 concurs and adds one should sit shiva, R Yaakov Ariel agrees with Kaddish but suggests saying it after Aleinu to distinguish between him and other orphans). He mentions rending his garments as a possibility but cautions against observing laws which compromise religious observance, e.g., refraining from Torah study.
I know a woman who is the daughter of a Jewish woman and non-Jewish man. When her father died, she was not obligated to sit shiva, etc., but her rabbi (a well known charedi rav) did call her to express condolences. So I don't think there is a specific halachic requirement, but the child could mourn as seems appropriate.