Impossible to answer from canon as far as I know. Moaning Myrtle's account certainly sounds like it is an instant effect.

However it is clearly not a spell in the way that Avada Kedavra is. (or we'd see green flashes flying out of its eyes constantly) I think it is something triggered in the victim when eye contact is made. A more subtle kind of magic perhaps. It seems like the brain decides to shut itself down at source.

It is probably trivially easy to stop being alive, we have just evolved to avoid doing so. The Basilisk's eyes could be compelling the brain to commit suicide while unlocking the, generally avoided, ability to do so. This has the bonus of being much more sinister than a generic killing spell, for me at least, as it uses the victim's own body against them.

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.

In my point of view, I suggest that the Basilisk’s sending a piece of its soul, which could be poisonous in nature or a human can not handle it in anyway, into a human through eye contact. Like how most witches and wizards use a wand to direct their magic, a Basilisk uses its eyes.

Or, it could be a swap in certain extremely small and resilient DNA strands that instantly make our own cells decay and rot which leads to us dying instant.

It honestly depends on opinion and what J.K Rowling tells us.

It’s not clear exactly how they work, but they cause instant death.

In Fantastic Beasts and Where to Find Them, it’s stated that looking directly into a basilisk’s eyes causes instant death. However, it’s not said how looking into the basilisk’s eyes kills.

“It has exceptionally venomous fangs but its most dangerous means of attack is the gaze of its large yellow eyes. Anyone looking directly into these will suffer instant death.”
_{^{- Fantastic Beasts and Where to Find Them}}

When Myrtle describes how she died, she said she saw yellow eyes, seized up, and then was floating away (so she was dead right after seizing up). It was very quick.

“So I unlocked the door, to tell him to go and use his own toilet, and then –’ Myrtle swelled importantly, her face shining, ‘I died.’

‘How?’ said Harry.

‘No idea,’ said Myrtle in hushed tones. ‘I just remember seeing a pair of great big yellow eyes. My whole body sort of seized up, and then I was floating away …’ She looked dreamily at Harry. ‘And then I came back again.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 16 (The Chamber of Secrets)}}

We don’t know what her body looked like, so we don’t know what damage looking into the basilisk’s eyes caused but ‘seizing up’ almost sounds like she’d been something like Petrified, but in a strong enough way to actually kill. This would fit with how people who see its eyes indirectly get Petrified but not killed, but there’s no confirmation that this actually is how it works.

I have a theory, not sure how plausible but here it is

In The Flash tv show there was a villain Rainbow Raider who could alter a person's brain's chemical activity to incite various emotions by flashing certain colors from his eyes. I think Basilisk too flashes certain colors which affects brain's chemical activity resulting in an immediate shut down of neural activity, which in turn shuts every organ in the body. Since brain has shut down, the person couldn't even feel pain. And the reason an indirect stare petrified but doesn't kill is because the colors get slightly distorted due to refraction of the light