Rambam writes in Hil. Talmud Torah (1:7) that one must pay to teach his son the entire written Torah.
וְחַיָּב לְלַמְּדוֹ בְּשָׂכָר עַד שֶׁיִּקְרָא תּוֹרָה שֶׁבִּכְתָב כֻּלָּהּ
Kessef Mishna (and others) understand that this is the only thing one must teach his son in accordance with the ruling of the Gemara (Kiddushin 30a).
כסף משנה: היה מנהג המדינה וכו'. (פ"ק דקדושין) [פ''ד דנדרים דף ל''ז] ומשמע לי דמכאן ואילך אין חייב ללמדו אפילו בחנם
Rambam writes in Sefer HaMitzvos (assei 11) that teaching and studying is considered one commandment of "v'shinantam l'vanecha".
היא שצונו ללמוד תורה וללמדה
Does he hold that the Biblical obligation to study Torah also only includes the Written Law?
Maharam Shik (quoted in Sefer HaMafteach) notes that given that the obligations to teach a son and a student are included as a single mitzva, one ought to be exempt from teaching the Oral Law to students, but he inst happy with that conclusion and he doesnt discuss ones personal obligation
I know that he writes (e.g. Hil. Talmud Torah 1:12) that one should divide his study and include the Oral Law in his study. I am also aware of what he writes in (1:10). The question is just whether study of the Oral Law fulfills the Biblical obligation. Bonus points for the views of other Rishonim (such as the Ramah quoted in the Tur)