Must the person to whom I swore by Al-Fatiha upon the Mus'haf absolve me so that I may be absolved, or can I absolve myself before Allah?

Reading only Surat Al-Fatihah while placing the hand on the Mushaf is not an oath, and nothing is entailed by it. Rather, it is a promise which is recommended to fulfill. There is no blame on you if you break it, as long as no harm results from that. However, if you swore by Allah or by one of His attributes not to do something, then you may do it if it is not forbidden, and an expiation for an oath becomes binding upon you.