There's a solid case to be made that the rules don't handle this case very well (in a way that matches what you'd expect from logic). Being petrified should pause biological processes like dying (if no damage is done to the stone).
So I'd rule that death saving throws are suspended by petrification, but the counter doesn't reset upon unpetrification and you aren't stabilized.
As an answer on
Can you be healed or stabilized (if downed) while petrified? points out, RAW you are still a creature (not an object) and only what it says in the Petrified condition applies, not anything more.
This technically means you can still heal from a short or long rest, which doesn't make sense to me if you aren't aging, and even your awareness of your surroundings is on pause (While petrified is the character aware? no).
You can be conscious and able to think without having any of your senses work, but my mental picture of petrified is not locked-in syndrome. Unless there's some reason to rule that petrified creatures are aware of the passage of time, it's easiest to consider them as "on pause".
So I'd propose this untested house rule:
- Petrified: you don't have to make a death saving throws at the start of your turn. (But the counter doesn't reset).
- You can't be stabilized by a medicine check while petrified (you're not biological anymore). Possibly allow some kind of engineering check instead, or allow getting bandages applied while petrified giving advantage when unpetrifying?
- You can't benefit from a short or long rest while petrified (time is on pause for your mind and bodily-healing processes)
Perhaps also disallow administering a healing (or other) potion to a petrified creature, but I don't know if that would be fun or not.
Taking damage while at 0HP would as usual cause you to fail a death saving throw (or 2 on crit).
These differences from the implications of RAW are more consistent with what we think of as being petrified, and hopefully flesh in this gap in the rules. (Pun intended, sorry.)