Unarmed attacks can crit, but do not do double damage

• Can an unarmed strike crit? - Yes

  If you roll a 20 on an unarmed strike, it is considered a critical hit.

  This means that it will automatically hit regardless of the opponent's AC and has the potential to cause 2 failed death saving throws from a creature at 0 HP.

  However, a critical hit does not get to increase its damage from a critical hit.

• Do unarmed strikes get double damage on a critical hit? - No

  Critical hits only double damage that you roll. Unarmed strikes do not have any damage rolled. So, RAW, critical hits do not increase the damage of unarmed strikes. See this Q&A for further discussion of this.

  Jeremy Crawford agrees and has (unofficially) clarified:

  An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

  This ruling, besides being in line with the literal reading of the RAW, is likely made to keep the rules of what damage to multiply from a critical (only rolled damage) consistent. If you have to start carving out exceptions for every static damage effect that doesn't roll damage but you think might kind of be like a roll, applying the critical hit rule could get complicated and cumbersome. Thus the simple, easy-to-apply ruling we get here.¹

Does the described attack cause failed Death Saving Throws? - No

• It is possible for an attack to do 0 damage

  Jeremy Crawford has (unofficially) clarified:

  There is not a damage minimum of 1, so it is possible to deal 0 damage with an attack.

  So you are indeed calculated the damage for that attack correctly.

• 0 damage does not count as damage

  Jeremy Crawford has (unofficially) clarified:

  Taking 0 damage is the same as taking no damage.

• No damage means no failed death saving throws

  If an Unconscious character takes damage while at 0 HP, they automatically fail one death saving throw, or 2 death saves if the damage is from a critical hit.

  If a creature takes damage while at 0 hp, then they will fail one saving throw (or two for a crit). If no damage is taken then no death saving throws are failed. The critical does not change this because damage is a prerequisite in order for any death saving throws to be failed in this way.

So the character in this case does not fail any death saving throws as a result of the attack

^{1 - Note that I've played in and run games where I've allowed unarmed strikes to double damage as a house rule, and it seems to have had only positive effects (if any, since it came up extremely rarely). But it was always made very clear that this exception was being made for only this one thing. If you did so at your table, I'd guess you wouldn't have any issues either. The only thing to watch out for is confusion I'd say, because the rule that only rolled damage is rerolled is a handy one, and breaking that for unarmed strikes might undermine that. I've only done this house rule with experienced players, so keep that in mind.}

Attack roll for unarmed strikes = d20 + proficiency bonus + Strength modifier.

Unarmed strikes are, in fact, melee weapon attacks - even though they're not made with a weapon. This is clarified in the Sage Advice Compendium:

What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon?

It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

Here’s a bit of wording minutia: we would write “melee-weapon attack” (with a hyphen) if we meant an attack with a melee weapon.

This is also supported by the fact that the rules on unarmed strikes are stated in the the "Melee Attacks" section of the rules; the wording is simply clarifying that if you want to make a melee weapon attack without a weapon, you can use an unarmed strike to do so.

As such, the attack roll is calculated as follows:

• d20 + proficiency bonus + Strength modifier