Because 'He does not have books' would be 'Han har inte bocker'. Inte- does not, Inga is more of a 'does not have "any"'. Hope this helped and made sense!
What's the difference between inge and inte? I believe this was answered in another question, but I don't know where.
Han har inga böcker = he has no books Han har inte böcker = he doesn't have books
That would be translated as "Han har inte böcker", rather than "Han har inga böcker", which would literally be translated as "He has no books".
Han har inte böcker would translate into He doesn't have books. I don't think this sentence is completely equivalent to the original.
'He hasn't any books' and I was corrected on my English? Apparently it's 'He hasn't got any books'.
We usually only use "have not" (or "haven't") for the auxiliary have in the perfect tense, not where have is the main verb. In that case you'd generally say "don't have" or "haven't got".
I mean people do say things like "I haven't any" but it's very informal and colloquial, and might be completely unacceptable to a lot of people. It seems to be getting more common though! And "have not" sometimes pops up in archaic or very formal language. The stuff in the first paragraph is the safest general rule