As usual, TB hit it right on the head. Fredo posed an inherent danger to the family and to the Family. Michael had NO CHOICE but to eliminate him.

While we may find Michael loathsome and cold, he did what he HAD to do.

As the end of the novel, Tom meets with Kay, and she asks him why Carlo couldn't be forgiven. Although Tom explains why, he also says, "In this world, he could have been forgiven." (I'm paraphrasing, btw). That also explains why Michael could never forgive Fredo's treachery, especially after seeing the depth of his envy.

Envy is a very dangerous emotion, and Michael recognized that.