'Ikke noget' is used when you say something like 'Nothing, it wasn't me that said it' and 'ingenting' is when you say 'There is nothing in the box'.

"Not anything" would be understood in English, but it's unnatural in most contexts (including this one). "There was not anything wrong" is one of the least awkward uses I can think of off the top of my head, but that still sounds pedantic compared to "There was nothing wrong" or "There wasn't anything wrong."