Indeed. Catelyn makes explicit why they’re considered abominations:
I think what you’re missing here is the connotation of “abomination,” which is religious in origin, whether we’re taking the original Latin meaning of an ill omen or how the word was used to translate from the Hebrew word “sheqets” which means ritually unclean or forbidden, taboo.
From this perspective, it doesn’t matter what Joffrey, Tommen, and Myrcella have done, it’s not a question of their culpability for an action, because in their very existence they are a violation of the laws of the Old Gods and the New. Hence why in the books the High Sparrow refused to give his blessing to Tommen without being “convinced” that the story isn’t true, because to do otherwise would be to taint the spiritual purity of the Sept of Baelor.
If the boy was truly Jaime’s seed, Robert would have put him to death along with his mother, and few would have condemned him. Bastards were common enough, but incest was a monstrous sin to both old gods and new, and the children of such wickedness were named abominations in sept and godswood alike.
The persons Joffrey, Myrcella, and Tommen were or have become are immaterial. The moment they were conceived, those three children were the products of a “monstrous sin” in the eyes of both faiths.
Is there anything in the books that reconciles the Church’s view on children being born of incest as abominations with the Targaryens long history of incestuous marriages?
The Revolt of the Faithful against “King Abomination” (King Aenys I) and its bloody repression.