Is the study of agadic material i.e stories which are not related to the practice or theory of Halakha, incumbent upon a Jew? In other words, is there something inherently wrong with not learning agadah as regards Torah study obligations?
The gemara (kiddushin 30a), when discussing a father's obligation to teach one's son Torah, says: (free translation)
"How far does the obligation to teach one's son Torah go? Rav Yehuda said in the name of Shmuel: like Zevulun ben Dan, whose grandfather taught him Mikra Mishna Talmud Halachos and Agados."
The gemara then discusses this further. But the whole discussion just revolves around how much is the father's (and grandfather's) obligation to teach the son, versus what the son himself needs to study.
This is quoted I believe in the Shulchan Aruv Harav (I don't have access to one right now to check, but I believe he poskens this.)
There is also the medrash on Mishlei- I think it's Yalkut Shimoni, perek 10- that says that each person will be asked in heaven if they studied mikrah, Mishnah, Talmud, agadah, and even more things like maaseh Merkava etc. If the answer is no, and the explanation is not acceptable, a person will be taken to task. (It's worth trying to see the actual text since it's very powerful. I'll try to find it later.)
You can also look in R' Dr. Yehuda Levi's work "Torah Study" where he has a whole chapter devoted to how the study of agadah is necessary to properly fulfill many obligations. i.e. mitzvos of bitachon, emunah, Simcha etc. can only really be understood well through study of agadah.