In the interest of looking deeper into this issue, I will do what I should have done in the OP and post Dante's argument, from Bk. II, ch. ix of De Monarchia. Please read it carefully, and if you cannot do that, then please refrain from contributing an opinion:
Furthermore whatever is acquired through trial by combat [duellum] is acquired by right. For wherever human judgment is unequal to the task, whether because it is wrapped in the darkness of ignorance or because no judge is available to preside, then to ensure that justice is not left abandoned we must have recourse to Him who so loved justice that, dying, he met its demands with his own blood; whence the psalm: 'The Lord is just and has loved just things.' Now this happens when by free agreement of both sides, not out of hatred, nor out of love, but solely out of a passionate concern for justice, we seek to know divine judgment through a clash of strength of both body and soul; we call this clash of strength trial by combat because originally it was devised as combat between two individuals. But just as in warfare all ways of reaching a resolution through negotiation must be tried first and only as a last resort do we engage in battle ... and just as in medical treatment everything must be tried before the knife and fire and these are to be used as a last resort; in the same way care must always be taken to ensure that, when all other ways have first been investigated as a way of resolving the dispute, we have recourse to this remedy as a last resort, forced to adopt it as it were by a need for justice. There are thus two identifying features of trial by combat: the first is the one we have just described; the other is the one we touched on earlier, i.e. that the contenders or champions enter the arena by mutual agreement, and not out of hatred, nor out of love, but solely out of a passionate concern for justice ... For if these essential conditions of trial by combat have been respected - and if they have not it would not be trial by combat - is it not true that those who out of a need for justice have come to confront one another by mutual agreement through a passionate concern for justice have come to confront one another in the name of God? And if so, is not God in their midst, since he himself promises us as much in the Gospel? And if God is present, is it not impious to think that justice can fail to triumph - that justice which he himself so loves, as we noted above? And if justice cannot fail to triumph in trial by combat, is it not true that what is acquired through trial by combat is acquired by right?
It does seem to me 'airtight', provided the conditions are respected. As I mentioned, the Catholic Encyclopedia entry on Ordeals is relevant:
Ordeals were a means of obtaining evidence by trials, through which, by the direct interposition of God, the guilt or innocence of an accused person was firmly established, in the event that the truth could not be proved by ordinary means. These trials owed their existence to the firm belief that an omniscient and just God would not permit an innocent person to be regarded as guilty and punished in consequence, but that He would intervene, by a miracle if necessary, to proclaim the truth. The ordeals were either imposed by the presiding judge, or chosen by the contesting parties themselves. It was expected that God, approving the act imposed or permitted by an authorized judge, would give a distinct manifestation of the truth to reveal the guilt or innocence of the accused. It was believed from these premises that an equitable judgment must surely result.
And further on, it discusses the trial by combat specifically:
The duel, called judicium Dei in the Book of Laws of the Burgundian King Gundobad (c. 500). (Mon. Germ. Hist., Leges, III, 537.) The outcome of the judicial duel was looked upon as the judgment of God. Only freemen were qualified to take part, and women and ecclesiastics were permitted to appoint substitutes. The duel originated in the pagan times of the Germanic peoples. In certain individual nations were to be found various usages and regulations regarding the manner in which the duel was to be conducted. The Church combatted the judicial duel; Nicholas I declared it to be an infringement of the law of God and of the laws of the Church ("Epist. ad Carolum Calvum", in Migne, P.L., CXIX, 1144), and several later popes spoke against it. Ecclesiastics were forbidden to take part in a duel either personally or through a substitute. Only English books of ritual of the later Middle Ages contain a formula for the blessing of the shield and the sword for use in the judicial duel; otherwise, no medieval Ritual contains prayers for these ordeals, a proof that they were not looked upon favourably by the Church.
It concludes thus:
However, a clearer recognition of the false ground for belief in ordeals, a more highly-developed judicial system, the fact that the innocent must be victims of the ordeal, the prohibitions of the popes and the synods, the refusal of the ecclesiastical authorities to cooperate in the carrying out of the sentence — all these causes worked together to bring about, during the course of the fourteenth and fifteenth centuries, the gradual discontinuance of the practice.
Unfortunately, the article contains no argument against the "false ground for belief in ordeals," which is disappointing. Has anyone run across, or perceives for himself, an explanation of this false ground?