Should the hotel have charged me for this?

Your safety was compromised here by a poor design. I see an OSHA violation here …

Well, he said it was in another country, so no OSHA. Plus, OSHA is for worker safety, not customer.

The hotel most likely had their insurance cover the damage too. There should had been a door stopper to keep the door from being able to hit the wall.
I would had said “No your insurance will cover it for you !”

First, check the bill to make sure that it does not say “tip for fitness center staff” instead of door replacement.

And you don’t understand the balance of power in these situations. The hosts are giving lots of business, by which I mean money, to this hotel. Lots more than the cost of a door. If something like this happened at a hotel where the conference I’m involved with uses, our Arrangements chair would be down there in two seconds saying “charging for a door that broke? Are you kidding me?”

Things in hotels gets lots of use. They wear out. Not your fault at all.

I definitely know nothing about norms for hotel use in developing countries, so I would like to amend my previous comments to reflect "In the US … "

They are using the wrong type of glass for their shower doors. All shower doors should be made from tempered glass that does shatter into miniscule little pieces. Large shards of glass breaking on a shower door are much more dangerous.

I agree. I assumed, unfairly, first-world.

When I first read the story I figured it was a hotel not in the United States. I think your chances of getting a refund are fairly small.