I agree with the premise of the question - common sense indicates that a petrified creature does not need to eat, drink, or sleep. I would even venture to say that this is likely RAI as well. But, the question asks whether this can be justified, RAW. So here's my attempt. (Incidentally, I also believe that @TimGrant's accepted answer is wrong1, because a creature's need to eat, drink, or sleep is typically found in its lore, not its stat block.)
Assume that time does not pass for the creature
The petrified condition says that a creature "ceases aging". What does this mean, especially in a game that has no listed mechanical effects of aging? Well, one thing it could mean (and the fundamental assumption of this answer) is that any process, or at least quasi-biological process, that requires time to have passed to take effect cannot affect the petrified creature. Things that have an 'accounting frequency' (damage taken each round, each turn, each hour, etc.) will happen to it, but anything that takes place after a certain amount of time as a threshold effect will not.
If we make this assumption, does the petrified condition then work like our intuition says it should?
Suppose I pour oil over the petrified creature and then light it. Flaming oil does 5 damage to a creature immediately when it enters the oil, or at the end of its turn if it was already there. Although there is a certain rate of damage (5 per turn), since the damage could be taken immediately it does not require a certain amount of time to take effect. At the 'end of the turn' is just when it happens for accounting, but the creature does not have to have spent an entire turn in the flames for the damage to take effect. Thus the petrified creature could be damaged by the oil (although it would have resistance).
Suppose I cast a feeblemind spell on the creature. The instantaneous effect is to force an Int save. Our petrified creature can still make Intelligence saves, so this happens. If it fails, its Intelligence and Charisma scores, which had remained what they were before petrification, now become 1. The spell permits a new save "At the end of every 30 days", but our creature will not be allowed this so long as it is petrified. Since time does not pass for it, it does not experience the 30 days and so cannot make the new save.
Suppose I cast sickening radiance on the creature and it fails its Con save, which it is allowed to make. The damage, exhaustion, and light all happen at the start of its turn, but they don't require that any specific amount of time has been spent in the radiance to take effect. The creature will take the damage, accumulate the exhaustion, and emit the light. Assuming the exhaustion doesn't kill it, that will go away at the end of the spell.
Now suppose the creature has been a day without water. The rules for dehydration say that the creature will suffer a level of exhaustion at the end of the day. But this assumes that the creature has passed the entire day without water. Unlike the 'end of the turn' example with oil, sufficient water at any point in the day would be enough. Here the end of the day is not just accounting for when the exhaustion happens, but is considering that a full day must have passed for this to take effect. Since biological time does not pass for the petrified character, they will not be exhausted from dehydration. Technically we could say they still need water, but cannot go without water for sufficient time for dehydration to have a consequence.
Similarly for food. "A character can go without food for a number of days equal to 3 + his or her Constitution modifier (minimum 1). At the end of each day beyond that limit, a character automatically suffers one level of exhaustion." Since the petrified creature does not pass the biological time required to reach the limit, they cannot be exhausted from starvation.
Finally, sleep. The PHB does not have rules for going without sleep, but suppose you use the optional rules in XGtE: "Whenever you end a 24-hour period without finishing a long rest, you must succeed on a DC 10 Constitution saving throw or suffer one level of exhaustion." In this case, the petrified creature cannot have the required 24 hour period pass for them; they cannot be exhausted from lack of sleep.
Making the assumption that time does not pass for petrified characters (at least biological time based on the fact that they do not age) permits them to be affected by damage and even imposed exhaustion, but not by things like dehydration, starvation, or lack of sleep.
1 It's a good faith effort, but it is wrong on two counts - a false analogy and an incorrect rules statement. The answer says that (1) when polymorphed, a creature assumes the stat block of its new form. This is true, but the creature here is being petrified, not polymorphed. That's the false analogy - nothing RAW says that the two processes are similar. Even if we accept this premise for the sake of argument, though, there is a more serious error. The answer says that (2) "A creature polymorphed into another form needs to eat what its new form eats. For example, a creature polymorphed into a Zombie gains Undead Nature and does not need to eat." This is simply false.
Eating (and sleeping) are not part of a creature's stat block. The Monster Manual (p.6) clearly states that the Statistics, or stat block, of a monster starts with its Size. Any descriptions that come before size are thus, by definition, not part of its Statistics. While there are several monsters with Undead Nature or Construct Nature that say they do no need to eat or sleep, these traits are listed before the stat block (in what is sometimes called the "Lore") and are not, properly speaking, part of their Statistics. Thus even if we accept the premise that petrification is like polymorph in that it replaces the stat block of the creature to that of the new form, such a change in and of itself does not mean that any lore-described Nature is assumed.
