My reasoning is simply asking “What Would Christ Do?”
And the answer is not even questionable. No one here has even claimed that this is a Christian act. That clearly proves that it is wrong.
Christ wouldn’t have cut off the head of a human being because somebody burned his favorite book.
I would observe that the Traditional Christian attitude toward Islam (cf. the critique written by St. John of Damascus, who understood Islam very well, having served as Grand Vizier to the Caliph of Damascus during one of those periods when Caliphs cared more about competence than Islamic piety in their court functionaries) is that it is a heresy, not a proper religion in its own right. I would also observe that it is traditional for Christians to burn heretical writings (cf. the treatment of the Acta of the Iconoclastic Synod of 754 by the Seventh Ecumenical Council).
Was it politic for an American to treat the Qu’ran in the way it deserves to be treated according to Christian tradition at a time when America is at war with some Muslims and allied with others? No.
As to “What would Jesus do?” Do you have the discernment to know that the content of the Qu’ran is not the sort of thing that would bring out the whips that drove the moneychangers from the Temple? It’s not so clear. Seeing that the Harps of the Spirit, as we Orthodox call the Fathers of the Ecumenical Councils, thought the Acta of the Synod of 754 should be reduced to ash, doing the same to an heretical text whose adherents have caused more suffering to the Christian people than all the iconoclasts before and after that synod might be exactly what Jesus would do.