Zev, technically, Henry VIII didn’t want a divorce; he wanted an anulment. The difference is that a divorce dissolves a valid marriage tie; an anulment declares that the marriage was invalid ab initio.
His argument was based on the canonical principle that a man should not marry his deceased brother’s wife, and Henry had done just that: Katherine of Aragon was the widow of his older brother, Prince Arthur, who died shortly after the marriage and never became King.
The Pope at the time of the marriage had granted Henry a dispensation from the canonical requirement, at the request of Henry’s father, Henry VII. However, Henry argued that the fact that his marriage to Katherine had not produced a male heir proved that the marriage was sinful, notwithstanding the dispensation, and the marriage should be annulled as void, because it was contrary to canon law. He relied in part on a passage in Leviticus about the seed not taking root in the brother’s wife. I’m going by memory here, so I may not be paraphasing accurately; I’ll see if I can find a cite. (The fact that he had the hots for Anne Boleyn, who was pregnant by him, was just a side issue.)
The Pope refused to set aside the dispensation, which contributed to Henry’s decision to separate the Church of England from the Church of Rome. However, doctrinally, the C. of E. under Henry remained very catholic in doctrine. The short version is that Henry replaced the Pope’s authority with his own, as Supreme Head of the Church of England. There were some changes towards Protestantism (e.g. - the dissolution of the monasteries), but for the most part the church was still catholic at the time Henry died.
Over the next two centuries, there was a considerable struggle over the doctrine of the Church of England - was it to remain catholic, but not Roman Catholic? was it to be Protestant along the Lutheran model? was it to be Presbyterian and Calvinist? was it to be disestablished?
Eventually, a middle of the road church evolved, with elements of both Protestantism and Catholicism. (One short hand is “Low Church” for those members who are more recognisably evangelical and Protestant, and “High Church” for those that look more Catholic - it’s more complicated than that, but the short-hand is useful.)
One of the doctrinal aspects where the Church looks more Catholic than Protestant is that it frowns on divorce. Different member churches of the Anglican communion take varying views on it, but the C. of E., to my understanding, is one of the stricter ones. Thus, having a Supreme Head marry a divorcee, and by strict doctrine thereby live in sin with her, caused considerable problems for the church hierarchy in the 30s.