Can you read German books, like Nietzsche's, by the end of the Duolingo course alone?
With a dictionary, as necessary.
Duolingo is focused on modern language. Nietzsche, on the other hand, used a language of more than a century ago, and his sentences are sometimes hard to understand even for native speakers.
In former times, authors (like Nietzsche or Thomas Mann or Franz Kafka) made much more complicated sentences than authors would write today. The beginning of Thomas Mann's "Dr Faustus" consists of two sentences comprising a whole paragraph of 21 lines.
Duolingo does not prepare you for the analysis of these types of sentences, I'm afraid. But it is a good basis, a starting point.
(Edit: I counted the introduction to Dr Faustus and gave the actual numbers)
No, not unless you laboriously looked up the majority of the words on each page. Even then, you'd probably run into some very confusing grammar. Personally I find it really interesting what texts I can understand and which are impenetrable... I can read some light news sites without a problem, but some literary texts are all but impenetrable because they have so much unfamiliar vocabulary. I've been watching Wagner DVDs with German subtitles lately--sometimes I understand everything in a given set of lines, but he uses a lot of obscure/literary/archaic language that often isn't in Google Translate's dictionary (which I use to look up words while I'm watching) and there are large sections where I can barely understand a single line on my own. I've studied a lot of vocab outside Duolingo, too! But even if I can't understand much of a text, I pick up on the wordplay and some of the literary qualities that I missed when I was relying on English translations, so I'm happy.
I actually tried doing something similar, in the days before Duolingo... Read most of an Andre Gide novel with a French-English dictionary and next to no background in French. Nobody had ever explained French grammar to me, so it was incredibly confusing! Took me ages to work through each sentence, and I had to write every word down on paper. Even then, I only understood about half of it (although luckily I had an English translation of the book to check my understanding against). I think I would have been able to understand if I'd had a combination of Duolingo and Google searches to help, but I'm not sure that really qualifies as "reading"... I couldn't get from the beginning of a sentence to the end without forgetting the meanings of several words. On the other hand, though, I will never forget the visceral feeling of the prose. And I had just graduated with an English degree at the height of the Great Recession, so I guess there were worse ways to spend my months of unemployment... :-P
As jsiehler remarked below, even if you can't understand the original text, it's really worthwhile to get a sense for the language. That's partly why I love opera so much--I can watch the same opera dozens of times, first with English subtitles and later with German ones, and pick up something of the flavor of the language even if I don't understand all the words.
Others have already said "no" and explained why not. But I will say - if you enjoy Nietzsche and want to read it, this is a case where a dual-language edition (with German on one page and an English translation on the facing page) might be very appropriate and enjoyable. It won't help your speaking any, but I think there's something to be said for going back and forth and getting a feel for the poetry of the language.
Darn. Obviously, I'm not deluded enough to think Duolingo will thrust me into the doors of academic German literature, but the barometer results with this thread really goes to show how hard it is to truly become learnt in a language. Whelp, off to German pronouns I go.
German seems thrillingly easy to me after being an expat in Asia for the past decade. Back in the US I found it so intimidating that I gave up, but now I'm blown away by how rapid my progress is and how quickly results come compared with Asian languages. I might not be able to read complicated German texts, but at least I can look at them and understand something of the literary quality of the writing, which is amazing! It's actually turned out to be one of the things I'm most grateful for about living abroad--I want to howl with laughter every time someone says that German is hard. Not that it makes German any easier for me, but it definitely lowers the intimidation factor and makes me conscious of how comparatively easy it is to attain real fluency. The discouragement of language learning can be so brutal. But at least it's really doable with a language like German--there are so many resources, and so many native speakers to help.
Aside from the points already made about old German and current German, generally you won't be reading books like that until third or fourth year college german, and I'm sure that the Duo courses don't take you to a third or fourth year college level. But if you do want to read books in German, I'd start with something easier, like the little prince...
Nietzsche, Schiller, Goethe, Mann, no. (To say nothing of Hegel or Kant). Simpler texts like Harry Potter, news articles, wikipedia articles, etc.: yes, with some difficulty at first. Depending on your vocabulary, the first chapter of Harry Potter will probably take you over an hour to get through, but after a couple of weeks at it it shouldn't be too challenging.
Think back to when you learned English. I remember being in 2nd grade and reading Harry Potter and loved it. The next year the first Lord of the Rings movie was coming out, so I tried reading the books, and couldn't make heads or tails of them - they were too far above my level at that point. It wasn't for another couple years that I tried taking another stab at them and found that I had enjoyed them. That is to say: it was 2 years more of practice and learning the language for me to get to a level where I could read and understand the books, and that's in an immersive environment where all I'm reading, hearing, and saying is the language of the book.
What you're asking after: Nietzsche represents some of the most difficult to read and understand texts in the entire German corpus even for German speakers. It's like finishing your Spanish program in high school and trying to dive into Don Quixote, or finishing American elementary school and trying to read Ulysses. You're setting yourself up for failure. Start with something easier and work your way up to those.