Hi everyone. I am new to the forum.

I think there was nothing that Fredo could do to make Michael forgive at that point.
First and foremost, Kay lost the boy (well, had the abortion) because of the shooting in Michael's mind. Second, the shooting endangered his children and Kay. I believe that FFC is genius with this plot- maybe this is the only believable reason to make Mike kill anybody.

Finally, just as an alternative if he lets him live even if he banishes him, Fredo would have been a constant threat to the family as he was the brother of Connie too.