Is thick fruit skin safe to be eaten?
this depends very much on:


the specific kind of fruit you're about to eat; 
Cultural normatives;
your personal flavor and preference;


For instance: I eat Avocados with the skin. Seriously. I do. Some of my friends consider it to be weird, but i couldn't care less - when they are ripe i just slice them as they are, stack 'em on a piece of bread and feast away with a sprinkle of salt. 

Most fruits i know of have edible skins - even citrus skin, as sour and zesty as it may be, is edible - and is used for producing all sorts of sweets, flavor extract and even food for animals.

I wouldn't go ahead and eat the skin of the Durian fruit - it's probably 'edible' but.... i donno.... it just doesn't feel right (plus the smell of it is UNBEARABLE)