The basilisk DOES kill by eye-to-eye contact. All the petrified victims don't meet the basilisk's gaze.
According to Greek mythology, the basilisk never had the power to petrify. There are no known accounts of someone looking at a basilisk's eyes indirectly in reflection or in any other way.
What the petrification reminds me of is Medusa. It seems Rowling incorporated that old legend into the Harry Potter saga.
Of course, doesn't really explain HOW the petrification worked. Maybe petrification is a watered down version of death? It can also be conjectured that petrification is a state which is closest to the state of being dead, without actually being dead.
Maybe the power would've been diluted even more if more reflections of the basilisk's eyes were involved.
EDIT: TO THE ACTUAL QUESTION.
From the book, we see that Mrs. Norris is petrified. In the Harry Potter universe, I don't really recall the souls of animals being present. Except horses? The Ghosts of Hogwarts? Supposing that animals DID NOT have souls, the severing the souls concept doesn't hold up.
Using purely my imagination here, a basilisk's eyes COULD vaporise the chi of a living being. The life force of a living being that is.
According to legend though, the only way to kill a basilisk is to make it look at its own reflection. It would then die of fright. From this, we COULD conjecture that its victim dies of fright. This doesn't reeaally gel with the petrification power though.