Objection 1. It would seem unlawful for the accused to escape judgment by appealing. The Apostle says (Romans 13:1): "Let every soul be subject to the higher powers." Now the accused by appealing refuses to be subject to a higher power, viz. the judge. Therefore he commits a sin.
Objection 2. Further, ordinary authority is more binding than that which we choose for ourselves. Now according to the Decretals (II, qu. vi, cap. A judicibus) it is unlawful to appeal from the judges chosen by common consent. Much less therefore is it lawful to appeal from ordinary judges.
Objection 3. Further, whatever is lawful once is always lawful. But it is not lawful to appeal after the tenth day [Can. Anteriorum, caus. ii, qu. 6, nor a third time on the same point [Can. Si autem, caus. ii, qu. 6. Therefore it would seem that an appeal is unlawful in itself.
On the contrary, Paul appealed to Caesar (Acts 25).
I answer that, There are two motives for which a man appeals. First through confidence in the justice of his cause, seeing that he is unjustly oppressed by the judge, and then it is lawful for him to appeal, because this is a prudent means of escape. Hence it is laid down (Decret. II, qu. vi, can. Omnis oppressus): "All those who are oppressed are free, if they so wish, to appeal to the judgment of the priests, and no man may stand in their way." Secondly, a man appeals in order to cause a delay, lest a just sentence be pronounced against him. This is to defend oneself calumniously, and is unlawful as stated above (Article 2). For he inflicts an injury both on the judge, whom he hinders in the exercise of his office, and on his adversary, whose justice he disturbs as far as he is able. Hence it is laid down (II, qu. vi, can. Omnino puniendus): "Without doubt a man should be punished if his appeal be declared unjust."
Reply to Objection 1. A man should submit to the lower authority in so far as the latter observes the order of the higher authority. If the lower authority departs from the order of the higher, we ought not to submit to it, for instance "if the proconsul order one thing and the emperor another," according to a gloss on Romans 13:2. Now when a judge oppresses anyone unjustly, in this respect he departs from the order of the higher authority, whereby he is obliged to judge justly. Hence it is lawful for a man who is oppressed unjustly, to have recourse to the authority of the higher power, by appealing either before or after sentence has been pronounced. And since it is to be presumed that there is no rectitude where true faith is lacking, it is unlawful for a Catholic to appeal to an unbelieving judge, according to Decretals II, qu. vi, can. Catholicus: "The Catholic who appeals to the decision of a judge of another faith shall be excommunicated, whether his case be just or unjust." Hence the Apostle also rebuked those who went to law before unbelievers (1 Corinthians 6:6).
Reply to Objection 2. It is due to a man's own fault or neglect that, of his own accord, he submits to the judgment of one in whose justice he has no confidence. Moreover it would seem to point to levity of mind for a man not to abide by what he has once approved of. Hence it is with reason that the law refuses us the faculty of appealing from the decision of judges of our own choice, who have no power save by virtue of the consent of the litigants. On the other hand the authority of an ordinary judge depends, not on the consent of those who are subject to his judgment, but on the authority of the king or prince who appointed him. Hence, as a remedy against his unjust oppression, the law allows one to have recourse to appeal, so that even if the judge be at the same time ordinary and chosen by the litigants, it is lawful to appeal from his decision, since seemingly his ordinary authority occasioned his being chosen as arbitrator. Nor is it to be imputed as a fault to the man who consented to his being arbitrator, without adverting to the fact that he was appointed ordinary judge by the prince.
Reply to Objection 3. The equity of the law so guards the interests of the one party that the other is not oppressed. Thus it allows ten days for appeal to be made, this being considered sufficient time for deliberating on the expediency of an appeal. If on the other hand there were no fixed time limit for appealing, the certainty of judgment would ever be in suspense, so that the other party would suffer an injury. The reason why it is not allowed to appeal a third time on the same point, is that it is not probable that the judges would fail to judge justly so many times.
Source: Summa Theologica (New Advent)