There is a fascinating book called "Watching the English" by Dr Kate Fox in which she explores the way the English use the word. I met my German OH in England and he was amused by the way we apologise for everything. When he came back to Germany, his friends remarked that he was apologising too much. I don'T say that it is always inappropriate, but we certainly use the word to cover a wide variety of situations that maybe devalue it when it is really needed.
If I bump into someone my mouth says sorry without the brain involved. If someone has died it's not just 'sorry' but I'm so sorry or really sorry or something with feeling added.
Sorry is derived from sorrow, expressing regret or sadness, so to that extent it is appropriate in a wide ranges of instances, ranging from the trivial to the very grave.
The Germans would say "entschuldigung" if they bump into you (well, they might if they are polite enough) and "es tut mir Leid" when something bad has happened. "Es tut mir sehr Leid" would be appropriate for a death or something really bad (sehr meaning very).