Can you die from your Constitution dropping and lowering your hit points?

Question

Let us say there's a level 5 fighter that has a 10 as his constitution score and 50 hit points. The fighter character gets into a fight and takes 40 points of damage, for 10 hp remaining.

Then let's say the fighter somehow loses 4 points of Constitution (for the sake of the question, imagine a DM's homebrewed poison that deals 4 points of Con damage/drain).

Now that the fighter's constitution score has dropped from 10 (+0 mod) to 6 (−2 mod), he would retroactively lose hit points based on his modifier times the current level: 2 × 5. His maximum hit point total would drop by 10 hit points and he only had 10 hit points left.

Would this situation drop the fighter to zero hit points because of the negative modifier to his constitution, or is there a rule that a character would have a minimum 1 hit point no matter what the constitution decrease is, until the character hits zero constitution and dies?

Answer 1

First: there is no such thing as ability damage in 5e. It is simply not a game term. There are not many official effects that even cause your scores to go down (related).

That said, if somehow your Constitution score does drop, the rules say:

If your Constitution modifier changes, your hit point maximum changes as well (PHB 177)

Thus what you fear would not happen. Only your hit point maximum changes with your Con modifier, but your current HP is not directly affected.

It is not explicitly stated as far as I am aware, but it is consistent with some official abilities out there that if your maximum HP drops to 0, you die. This could happen as a result of Con modifier change if you rolled for your HP increases and were unlucky and/or your hit die is d8 or below and you drop to 1 Con (-5 modifier).

Answer 2

Constitution damage isn't a thing in D&D 5e; Hit Point Drain is

The Wraith, Specter, and Wight have a life drain ability that, on each hit, reduces the target creature's maximum hit points by the amount of damage done. (If a Constitution saving throw is missed; DC 14, 10, and 13 respectively). Once the max hit points are reduced to zero, the target dies. (MM p. 302, 279, 300) This life drain ability appears to be this edition's version of the Constitution damage from previous editions.

The Shadow when it hits will reduce a target's Strength by 1d4 for each hit, causing death when Strength is reduced to zero. That's an ability score damage feature similar to what you are asking about. (MM, p. 269)

An Intellect Devourer does something similar by targeting intelligence, but it doesn't kill the target the way that the Shadow et al do.

Constitution is not a target of damage in this edition. (Based on the published books so far: PHB, MM, DMG, Sword Coast Adventurer's Guide, Volo's Guide to Monsters, Xanathar's Guide to Everything, Mordenkainen's Tome of Foes, and Tasha's Cauldron of Everything).