At the end of this answer to a different question ("If you succeed on all of your death saving throws, what happens?"), the answer states that:
If an unconscious character is attacked, the attacker will have advantage on the attack (if melee) from the Prone condition; if it hits, it will be an automatic critical hit because of the unconscious condition which is 2 death save failures.
I just wanted to clarify exactly what happens in this scenario; let's say the scenario in this case is a creature is attacking a downed but stable PC. In both cases below, assume that the PC just made their third successful death saving throw and is now stable but still unconscious, and it's now the monster's turn.
Is it:
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- PC is unconscious and monster attacks and hits;
- PC takes damage and is now back to making death saving throws;
- since the PC just took damage (and a critical hit, no less, as per Unconscious condition rules), they immediately suffer two death saving throw failures;
- on this PC's next turn (assuming there is no intervention by allies), when they roll a death saving throw, they may die if they roll below 10.
Or:
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- PC is unconscious and monster attacks and hits;
- PC takes damage and is now back to making death saving throws;
- the damage taken is what put them back on death saving throws, so they do not immediately take two failed saves;
- on this PC's next turn (assuming there is no intervention by allies), when they roll a death saving throw, they are currently on no fails and no successes, so they have no chance of dying this turn.
Best Answer
The creature takes death saving throw failures on the initial hit
Unfortunately for the character in question, your first scenario is correct.
A stable creature is unconscious with 0 hp - the fact that they are currently exempt from having to make death saving throws (and that the number of successes/failures they have previously made has been reset) is irrelevant.
The rules for death saving throws state:
So, as a stable creature has 0 hp, any damage it suffers causes at least one immediate failed death save.
Damage should still be rolled to check for Massive Damage, and if the attacker is only 5 feet away from the character, it is an automatic critical, meaning two failed death saves.
Then, further compounding the character's misery, on its next turn, it must also resume making death saving throws.