I'm no Godfather trilogy expert. I have seen Godfather and GF 2 many times so I guess I can offer a female's point of view. It's about POWER, not violence imo.

Michael betrayed Kay. Fact. By Kay having an abortion, she had complete power over him. He needs to take the power back.

(My opinion of the abortion scene and the closing door scene )

Kay represents all his weaknesses, his failures. He cannot even keep his wife happy nor cares to know anything negative about his children. Michael is a coward who cannot cope in the NON-Mafia world. (He is not even 1/4 of the man his father was in that respect) Vito was powerful as a father and a don. And YES eventually, Michael would have verbally and physically abused Kay.

Kay still has some power over Michael because she took something from him, a son. I always say that there's no such thing as "forgive and forget" in the Italian male DNA. It shouldn't be that way with your wife, your partner, the one who you gave your heart to. He should have worked things out but he was too weak and he would also be admitting he was wrong, that he BETRAYED his own wife. Instead, Michael needs to show his wife AND KIDS that no matter what he does RIGHT or WRONG, he is ALWAYS RIGHT. He will show his ultimate power. So when he closes the door on Kay, he is well aware of the effect it could have on his kids. He needs to convince himself and show his children that he is the one who has been betrayed. Kay went against BOTH families. He needs to show that he is powerful in both of his worlds: His own family and his mafia famiglia. He needs to take back the power Kay once had over him.

(For those of you who are GF buffs please go easy on me, like I said, I'm no expert on these films. Thnx)