Why didn't the Basilisk kill its victims after they were petrified?

It would probably be quite difficult to kill a petrified person, and doing so might not be sensible.

The word “petrify” is usually used to mean “turn to stone” (originally from the Greek word petra, meaning rock). Although I don’t think the condition is explicitly defined in the Harry Potter canon, we have no reason to think the definition is different from convention.

(My answer to Besides the Basilisk, what kind of magic can cause petrification? has some fairly flimsy canon about Gorgons, which link petrification and turning somebody to stone. That’s the best I can think of.)

It’s probably quite hard to kill a person who’s been petrified, if they have truly been turned to stone. There would need to be a link between the life of the original person and the status of the stone.

About the only thing I can think of would be to damage or destroy the stone, but if there was a way to repair the stone before reconstituting the person, then it would probably be for naught. If you were thorough, then you could grind it into dust and scatter the dust to the four corners of the earth, but that’s beyond a Basilisk’s capabilities.

I’m sure the Basilisk could smash a statue (perhaps by throwing it against a wall), but it might not be sensible to do so. Not unless it could cart the person off first, but that would detract from the intimidation aspect.

That would cause noise and attract unwanted attention. It might also make it easier to deduce that it was a Basilisk at work. The class of creatures which can petrify somebody is sufficiently large that a Basilisk isn’t the only candidate; but attacking the petrified person would narrow the pool. Part of the Basilisk’s advantage is the element of surprise; giving clues to its identity would be unwise.

There is no canon answer to this question, although it is a great one! This issue is not addressed in the main Harry Potter novels or any of the ancillary books or at Pottermore.

It's possible the Basilisk didn't know the intricacies of its own powers. It was, after a serpent created from a chicken's egg and a toad -- not exactly Einstein material. It's possible, when it saw its victim topple over seemingly dead, that perhaps it thought it had succeeded in causing death and merely slithered away, not knowing the victim was lying petrified on the floor, fully alive.

Why did it not eat its victims? I don't know -- canon doesn't say. However, we do know that the Basilisk took orders from Tom Riddle, who ordered it to kill Muggleborns. Riddle wanted to be known as the Heir of Slytherin. If students had simply disappeared (i.e. killed and then eaten), Dumbledore would have never recognized that students et al were being petrified, which indicated a Basilisk, which in turn eventually led to Tom Riddle and his heritage as Salazar Slytherin's one true heir. Riddle may have ordered the Basilisk to kill; the Basilisk may have obeyed this command, but had other thoughts along the way, like, Man, I'm hungry ... It's been yonks since I've had a nice fresh first-year ...

There is no logical explanation for why Basilisk did not eat its victims. Snakes are known to swallow their prey without any problems while they are still alive. If Basilisk is a snake, and if he is hungry as he says, then nothing would stand in his way of eating the students. Except for the fact that Harry Potter is a children's book, and what kind of children's book would it be in which children would be killed and eaten by a giant snake. So my opinion is that JKR didn’t really think too much about this. As for the word "petrify", in this context it means "to stun", to freeze the victims in place, not to turn them to stone. Let us remember that Draco used the spell of Petrificus Totalus on Harry in the train, and after that he broke his nose with a kick to the head. If Draco could break Harry's nose, then Basilisk could tear his victims apart, regardless they being under the influence of what it would seem the same spell. It remains unclear whether Basilisk's glare has more power of petrification over the spell of Petrificus Totalus.