Call API on post save/update and show the result in admin area

You can’t do that with PHP. You must use Javascript-React to alter/enhance block editor behaviour.

You’ll have to use subscribe function, for subscribing to actions, catch isSavingPost action there (there is also isAutoSavingPost for autosaves) and dispatch a Notice in the editor with appropriate action. Familiarity with React and Redux would be of great help.