Yes, you are right re "Henery the Eighth, I am".
I found a youtube clip
which shows Patrick Swayze singing Henry the Eighth to Whoopi Goldberg, as you say.
Ghost is one of my all-time favourite movies, and I've seen it many times, but don't own a copy.
It's been a while, but I remember a later street scene, where Whoopi tries to renege on going
downtown and Patrick responds by singing 99 bottles of beer.
Whoopi then quickly changes her mind and goes downtown as agreed.
I could be mistaken though, and can't find a supporting youtube clip.
Update: Found a supporting youtube clip.