Many of the most profound written works on the subject of cultural and generational clashes are true accounts; these often come in the form of autobiography or memoir, some of which are listed below.
One book which fits your guidelines—cultural conflicts and generation gap—is an autobiography by James McBride entitled The Color of Water: A Black Man’s Tribute to his White Mother. This story is set in the United States, but it explores the conflicts between several cultures, including white and black, Christian and Jew. It is written, in alternating chapters, by McBride and his mother. The story documents McBride’s struggles with his mother’s denial of her cultural heritage as well as her generational beliefs. The two of them are from distinctly different generations, which is one other cause of their conflict.
Another suggestion is Annie John by Jamaica Kincaid. This is also a rather autobiographical novel about a young girl who develops distinctly different ideas about what defines happiness and success. It is set in Antigua, in the West Indies, when it is a British colony; and the novel discusses the effects of colonialism on a culture.
When I Was Puerto Rican by Esmeralda Santiago is another autobiographical account of the author’s struggle to forge her own cultural identity both as a Puerto Rican and as an American. It also contains the generational issues between her parents' more formal beliefs and the more modern beliefs of her generation in America.
The Kite Runner by Khaled Hosseini is not just about these two issues, of course, but it certainly contains a lot of cultural conflict once Baba and Amir move to the United States as well as the generational conflicts which separate Amir and his father.
Adeline Yen Mah’s memoir is entitled Falling Leaves: The Memoir of an Unwanted Chinese Daughter, and it tells the story of her constant battles with her father and stepmother while living in a country full of conflict and oppression. Another version of this story is a revised version entitled Chinese Cinderella: The Secret Story of an Unwanted Daughter.
Another idea is Laura Esquivel’s Like Water for Chocolate. The protagonist strives for her own identity in a country which is in the midst of a revolution of country-wide change and in a family tied to traditions that no longer seem relevant.
The Girl in the Tangerine Scarf by Mohja Kahf is another novel in which the protagonist, Khadra Shamy, struggles to find her cultural identity as well as separate herself from her parents’ narrow-minded, often racist, and certainly not always correct beliefs about Americans.
One time her father told her Shakespeare was really an Arab. “Just look at his name: It's an Anglicization of Sheikh Zubayr,” he said with a straight face.
Clearly this kind of thinking is traditional versus modern, and it is bound to cause some friction and division between a traditional father and daughter who is trying to establish her own personal and cultural identity.
The classic The Good Earth by Pearl S. Buck is another story about generational differences and cultural conflicts. You mentioned Diane Abu-Jaber, and certainly her Crescent: A Novel would fit into this category, though of course it concentrates more on the cultural conflicts than the generation gap.
There are plenty of other choices for stories which suit your guidelines, of course, so take this as a mere starting point for discovering the kind of text for which you are looking.
Tomorrow when the war began (John Marsden) and possibly the Ellie Chronicles (which are a follow-up series to that), dealing both with generational and cultural gaps or conflicts.
Murder in Memoriam (Didier Daeninckx) is both and is set against the backdrop of France over certain years.