Just read the "Resting" rules.
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds. (PHB p.186, "Short Rest")
We stipulate that you spend at least an hour. The question, then, remains whether being unconscious at 0HP is "more strenuous" than eating, drinking, reading, tending to wounds.
No, it's not more strenuous. An unconscious character, by rule, can neither take actions nor move. (PHB pp.291-292, "Appendix A: Conditions") Eating, drinking, reading, tending to wounds are actions--no, not in the combat-economy sense, but in the plain-language sense.*
Perhaps, though, the mere state of being at 0HP, struggling autonomously to stay alive while mortally wounded is more strenuous? But you're not struggling to stay alive, you're stable; you're not in mortal danger and you're resting so calmly that in d4 hours you'll have regained a hit point. (PHB pp.197-198)
Having met all the prerequisites of a short rest, you gain the benefits of a short rest. This includes, per "Short Rest," the ability to regain hit points through the expenditure of Hit Dice.
But "Short Rest" says the character spends Hit Dice. To do that doesn't the character need to be conscious? It doesn't say so. If you can tell me what it looks like when a character spends a Hit Die--in game!--then perhaps we'd have to conclude that the character can't do it while unconscious. But Hit Dice are a metagame concept, methinks, and it's strange to ascribe their operation to the character.
Something being "strange" to me shouldn't sway you. But this should: contrast the requirements for a short rest with those for a long rest. The long rest has similar time and activity requirements described. Then there's this:
A character must have at least 1 hit point at the start of the [long] rest to gain its benefits. (PHB p.186, "Long Rest")
The designers were clearly thinking of exactly this situation, for when else is a character at 0HP? And they explicitly state that 0HP can't be the start of a long rest, but they conspicuously don't state that 0HP can't be the start of a short rest.
"Doesn't that moot the "regain 1HP in 1d4 hours clause?" Nope. Imagine a character with no HD to spend--they've got to wait the full d4 hours. Or a player could choose not to spend the HD and, instead, save them for later.
* - see also every time you've been sleeping vs. every time you've been awake.