Can a son of a kohen that married a non-observant giyores be called up to the Torah for the first Aliya?
As @Deuteronomy and others have commented, see here, Shulchan Aruch HaRav
Any child who was born from parents who were in a Biblically Kehuna prohibited relationship, whether within wedlock or out of wedlock, is defined as a Challal. Meaning, that if a man who is a Kohen has a child with a woman to whom he is Biblically prohibited in marrying due to his status of priesthood, then any offspring born from that union is defined as a Challal. For example, if a Kohen had a child with a woman who is Halachically defined as a Zoneh...."
A gentile is considered a Zonah.
So, he is not normally given the first aliyah.