So I was playing against High Elves, and he has a hero on an eagle. I shot him with my Hail of Doom, doing no damage to the rider, however, I killed the eagle. Does my opponent have to make a panic check for the hero?

My logic is that the unit once consisted of a mount and rider, and thus the mount's wounds would count towards the total unit. Am I correct in assuming this?

It's a character on a mount, it's not a unit with two equal parts. No panic.

Exactly.
If you want a BR reference, is in the "Panic" paragraph, in the Psychology section.

Thanks much!