The PHB says a short rest is a minimum of 1 hour but doesn't say anything about a maximum duration. My players argue that because it only takes one hour, they can short rest every hour. This worries me because a fighter can heal up using Second Wind, short rest, use Second Wind again, and short rest again, indefinitely until their HP is back to full. That doesn't sound right to me, is there anything in the rules that prevent this?
I'm basically asking if the characters rest for two hours, does it count as two short rests or one? This rule seems ambivalent as to who decides if the short rest is longer.
Additional Background:
A player left the game when I invoked Rule 0 because he said I was being unfair when I offered to roll 1d6 for how many hours a short rest lasts. This rule is ambiguous and caused so much trouble in our game with regard to the Warlock and Fighter and other classes to a lesser extent.
I argued a fighter's second wind could only be used during a fight to reinvigorate his battle capabilities, but the fighter insists as written he can rest every hour and use it as a form of free healing. Similarly, the Warlock can use his sleep spell on an encounter, leave and come back an hour later and use it again. The wizard get to regain a few of their cast spells an hour later as well. So, everyone goes into a dungeon, blasts things, then runs out, rests an hour or two then heads back at full or near to full capacity.
Best Answer
The rulings
My previous answer for this part was based on my reading on it. Crawford has talked, though.
and
The first one would still be subject to "I can simply break the 4 hours by casting cantrips in the air for a few minutes", so it would not be 4 "unbroken" hours any more. The second statement is stronger though: it says you, oficially (i.e. not Rule 0), decide what counts as actually breaking a rest and starting another.
The World
Even if the rules don't prevent your fighter and warlock from short resting for many consecutive hours, there are lots of things preventing your scenario from happening in the adventure.
Where exactly are these characters? They can just stay still, unmoved, for three hours, and nothing happens? Why aren't they being swarmed by enemies?
That's how I usually deal (and how other DMs I've played with and have watched deal) with rests: you can take a short rest and then start moving before every enemy nearby decides to eat you. In your specific scenario (overworld, as mentioned in a comment): are they on the woods? Make a hoard of giant spiders notice them after one and a half hour. Are they in a plain field? Oh, what is that flying over there? Is it a Young Green Dragon looking for food near its lair? Make them feel uncomfortable staying still for hours in the same dangerous place.
Honestly, if they can just sit still and relax for 3 hours uninterrupted, they might as well just sleep and relax for 8 and make a long rest, then everyone, not only the fighter, is back to full health, spell slots and class features.
But this usually isn't the case. Remember, the world is alive (I've read this phrase often here, and I liked it), if they are wasting time resting every hour each 1 hour and 10 minutes, they are wasting time. Conflicts are happening and resolving and they are being left apart because they are singing around a firepit. Maybe they missed a quest. Maybe a friend has died.
The Final Word
If you really want to hard rule it, remember you are the DM. In the end, you decide everything, from how long a short rest is to how many times Second Wind can be used in a given time interval. The book is your tool, not your master. But my point is that even if you want to go RAW and RAW ends up allowing this "exploit", you can still prevent it from happening in a number of ways without any extra ruling. Solving it without hard ruling will make the players feel less frustrated (as you mentioned one player has already left because it felt unfair that he was restrained by this method) and it is actually intended, as the world is dangerous and you can't just sit and rest anywhere in a dangerous world.