It would be killed.
My dictionary defines “fatal” as “causing death” – it would be killed by the crowing of the rooster, not just petrified.
If a semantic argument isn’t enough, we can look to mythology. Much of the lore around magical creatures in Harry Potter comes from classical myths; the basilisk is no exception. In Bulfinch’s Mythology, the passage on the basilisk reports the same weakness:
There is an old saying that “everything has its enemy” – and the cockatrice quailed before the weasel. The basilisk might look daggers, the weasel cared not, but advanced boldly to the conflict. When bitten, the weasel retired for a moment to eat some rue, which was the only plant the basilisks could not wither, returned with renewed strength and soundness to the charge, and never left the enemy till he was stretched dead on the plain. The monster, too, as if conscious of the irregular way in which he came into the world, was supposed to have a great antipathy to a cock; and well he might, for as soon as he heard the cock crow he expired.
(The basilisk and the cockatrice are not always considered to be the same animal, but Bulfinch treats them as the same. In this version of the myth, the basilisk is created by hatching a rooster’s egg under a toad or serpent, hence “irregular way in which he came into the world”.)
And Bulfinch is just a collection of existing myths; we can find this going back even further, to a passage from Aelian’s Characteristics of Animals (2nd century AD):
With its crowing a cock scares a lion and is fatal to a basilisk.
The exact phrase in the original Greek is “ἀναιρεῖ δὲ βασιλίσκον”. The third word is Basiliskon, and the first word, anairei translates as “to kill” or “to put to death”. There can be no doubt about the meaning of this sentence.