Would Tywin Have Executed Tyrion?

Oberyn Martell was Tyrion's champion and Gregor Glegane was Cersei, the accuser's champion. As Oberyn died, Tyrion was sentenced to death, so technically, Tywin has not personally decided to kill Tyrion.

Would Tywin have actually executed Tyrion?

Originally Answered: Would Tywin really have let Tyrion die? No. But not because he cares for Tyrion, he doesn't. However a public execution of a son of his is a pretty big slight to his house and we all know how Tywin feels about those.

Did Tywin like Tyrion at all?

Tywin hated Tyrion, but he respected him. He respected his cunning, intelligence, and political acumen. The fact that he respected Tyrion probably made him hate him more.