Is repentance obligatory, and how should it be done, after the Qur'an accidentally fell to the ground, and the feeling of guilt still persists?

The falling of the Mushaf (copy of the Qur'an) without your intention is not considered a sin, because Allah has lifted accountability from the one who forgets and the one who errs. Therefore, no sin befalls you in this, nor is it considered a shortcoming on your part. So, beware of obsessive whispers (waswasah).