A player party is battling multiple enemies, one of which has been rendered unconscious by the Sleep spell. The PC Bard decides to comedically incapacitate the other enemies by casting Stinking Cloud on them, but the unconscious enemy also happens to be in this spell's area of effect. The DM judges that the sleeping enemy (failing a CON save) would be awakened by the act of choking from the gas, though they would also be incapacitated for a round.
According to the stinking cloud spell description in the 5e SRD:
On a failed save, the creature spends its action that turn retching and reeling.
The description of the sleep spell indicates that it remains in effect
until the spell ends, the sleeper takes damage, or someone uses an action to shake or slap the sleeper awake.
However, while the stinking cloud spell is treated as "poison" for the purposes of the saving throw, it does not directly cause damage, nor has someone directly shaken or slapped the affected character.
Has the DM made an incorrect ruling on the interaction of these spell effects, based on a strict interpretation of the rules for removing sleep's effects?
By RAW he is incorrect, although I do think you can infer that RAI he's probably in the right, I'll discuss below.
Sleep states the two conditions which end it. Neither of these conditions are met by retching and coughing while asleep. The unconscious creature would be asleep but also coughing and retching. If you will, picture somebody who's passed out because they're severely intoxicated. Choking does not automatically wake you up, and can in fact be entirely fatal.
In fact, you can by RAW completely submerge a sleeping creature in water, at which point they will begin drowning until the spell wears off. Since they haven't taken damage or been shaken/slapped awake, submerging them would not automatically wake them.
As a DM I am looser and look more to RAI instead of strict RAW. The shaken/slapped awake portion leads me to believe the intent is that normal, direct methods of awakening a deeply sleeping individual would work, thus I would not permit submerging a sleeping person to not awaken them (since you will definitely be awakened by a bucket of water in the face). However by RAW, they are still asleep.