By that logic, then, a Statued creature would not be turned to stone by the basilisk (as they are already stone); they would only fall unconscious. After recovering from being unconscious, or at the expiration of the Statue spell, the creature would then be able to revert back to flesh.
To me, the bigger question is whether a creature would count as a construct for the purposes of the Statue spell. (A construct is immune to all effects that would require a Fortitude save, unless the effect also works on objects or is harmless. Though, that would create a debate on whether or not a gaze attack such as this is "harmless"...). It doesn't indicate such in the spell description, but the material the creature is made of would be similar. While in statue form, the creature is "made of solid stone", implying that it cannot move while in that state; alternately, a construct can move by its own propulsion, while still being made of a substance such as stone or metal.
As it stands, since the creature is already made of stone, if I were DMing the game I would rule that the player under the Statue spell remains conscious and "alert" but unable to transform back to their normal form without a Stone to Flesh spell, even after the Statue spell expires.