How can one be absolved from the vow "that I will not put anything on my face unless I consult a doctor" after having placed a hand on the Mushaf?

Consulting a doctor regarding permissible matters is neither an act of drawing closer to God (qurbah) nor an act of obedience (ta'ah). There are two views concerning vows related to permissible acts: The first is that one has a choice between fulfilling the act without expiation or abandoning it with an expiation for breaking an oath (kaffarat yamin). The second view – which is the more preponderant opinion and that of the majority of scholars – is that such a vow is not valid and requires no expiation. Therefore, you are not bound by anything in this vow. However, if you offer an expiation for breaking an oath as a precaution and to avoid scholarly disagreement, it would be better.