There is tons of info on his many personal motivations for the split, including securing a dynasty and personal lust/love, as well as many other factors. But I can’t find anything about how he justified the action in a Catholic/Christian framework. Did he even bother?
Henry VIII was not especially enthusiastic about the continental Reformation, and therefore the royal Reformation under Henry did not appeal to quite the same arguments. Henry did not argue that the Roman church erred on important matters of doctrine because he thought the Roman Catholic church was basically right on most of the doctrinal points.
Henry's arguments for separating the English church from Rome rested on two key points. The first was that the papal dispensation he received to Mary Catherine of Aragon in the first place was invalid and never should have been granted. Catherine had previously been married to Henry's brother Arthur prior to his death, and it was unclear if the marriage had ever been consummated. Under such circumstances, they could have been marrying under prohibited degrees, and would require special permission from Rome. Henry and his advocates argued based on passages from the Old Testament that seemed to suggest that not having sons was a potential punishment for incestuous marriage that the marriage was invalid and should be annulled. Henry's reasoning on this was doubtless colored by his own motivations at that point, but he does seem to have actually believed this at some point.
The second major point was the belief that English kings held a rank equal to an emperor. What Henry is really doing here is jumping into a centuries-old dispute about the power of temporal rulers of religious affairs. There had been those throughout late antiquity and the middle ages who held that emperors had a unique authority over religious matters within their own realms that ordinary kings lacked. It was, after all, Roman Emperors who had called into session the great councils of the undivided church. If Henry VIII were an emperor, he would stand on stronger ground in overruling the Pope on religious matters within his own territory. The 1533 Statute in Restraint of Appeals reads, in part:
"it is manifestly declared and expressed that this realm of England is an Empire, and so hath been accepted in the world, governed by one Supreme Head and King having the dignity and royal estate of the imperial Crown of the same, unto whom a body politic compact of all sorts and degrees of people divided in terms and by names of Spirituality and Temporalty*, be bounden and owe to bear next to God a natural and humble obedience: he being also institute and furnished, by the goodness and sufferance of Almighty God, with plenary, whole, and entire power, pre-eminence, authority"*
Note that Henry is claiming, as an emperor, both spiritual and temporal authority, not just temporal power. The 1534 Act of Supremacy likewise reads, in part:
"the king, our sovereign lord, his heirs and successors, kings of this realm, shall be taken, accepted, and reputed the only supreme head in earth of the Church of England, called Anglicana Ecclesia; and shall have and enjoy, annexed and united to the imperial crown of this realm, as well the title and style thereof, as all honors, dignities, preeminences, jurisdictions, privileges, authorities, immunities, profits, and commodities"
Henry VIII was definitely working for his own self-interest in these debates, but the arguments he used were not pulled out of thin air, and people of his time understood the vocabulary he used even if they thought he reached the wrong conclusion.
For a general history of the English Reformation, see Christopher Haigh's English Reformations