Could people in the medieval ages get divorced?
Annulment and divorce what quite common, not 50% like it is today, but still quite common. An annulment was a declaration that the marriage is invalid. This may come from a variety of reasons. Maybe someone was pressured into the marriage, that person can talk to a panel of clergymen and if they decide that the person was pressured into the marriage, the clergymen would rule that the marriage never took place to begin with. The the famous renaissance example, Henry VIII. He claimed that because Katherine of Aragon was previously his brother's wife, his marriage should be annulled. Under most circumstances an annulment would be granted. However, Katherine was the aunt of the Holy Roman Emperor and he pressured the Pope not to grant the annulment. His troops actually invaded Rome. Now divorce was different. A divorce is a personal and legal separation between man and wife after the two made an oath before God that they would be together "till death do they part." You were allowed to get divorced, but the Church wouldn't recognize it and they wouldn't allow you to get remarried. So in the Middle Ages it was stupid to get a divorce because having legitimate children is pretty much essential, so if you wanted to split from your spouse you had to come up with a reason why the marriage shouldn't count in the first place. But they were granted quite often, we just have one notable exception.
Why did king Henry VIII want a divorce?
He didn't. In fact, divorce was illegal, so, he could not have gotten one even if he did want to. When his older brother died, his father, Henry VII, had Henry marry his brother's widow, Catherine of Aragon. This was actually an illegal marriage, and against canon (church) law, and needed Papal dispensation. Henry saw the lack of a male heir as God's punishment for his illegal marriage. He was a devout man, and took this very seriously. It has to be said, that he was also interested in Anne Boleyn at the time too, but... that was not the main thing. At first. He sought to annul the marriage, which is the way the Church recognizes that the marriage was not proper in the first place, and therefore never actually happened. However, Catherine's relative, Charles V, the Holy Roman Emperor was holding the Pope hostage at the time. So, the Pope stalled, when normally granting the annulment would have been fairly routine. He kept on stalling. Meanwhile, the then Archbishop of Canterbury died. The king, as Head of the Church (which English kings had been for 100's of years) would appoint bishops, with the co-operation of the Pope. The Pope would send the name on the "pilum," literally a stick with a message wrapped around it. Then, the king would make the appointment. But, Henry made his own appointment without the Pope's help. And, he appointed the Boleyn's chaplain. The new Archbishop looked into the matter (for some time), and declared the original marriage illegal, thus leaving Henry free to marry Anne, which he did. Mayhem ensued! The Pope was angry, and eventually excommunicated Henry, and, it all went downhill from there. There were other issues, like the Protestant Reformation in Europe (which Henry vigorously resisted), and the Peter's Pence, a tax to Rome that was being used to fund England's enemies, and was illegal under English law (as it made Henry vassal to the Pope), and the fact that the church owned about 1/3 of England, and was increasing a lot, and that land was exempt from taxes, and the people on it did not have to serve in the military. So, there was a LOT of stuff going on! Short answer: Henry wanted an annulment of an illegal marriage. NOT a divorce.
What if Henry VIII had not divorced Katherine of Aragon?
It depends on why. Does Henry VIII not divorce Katherine of Aragon because she bears him a living son or sons, or does he not divorce her because she predeceases him or he predeceases her, dying much earlier than in OTL? You've had some answers related to the latter, so I'm going to take the former and run with it.A living son born to Katherine of Aragon absolutely ensures her position as Queen and is welcomed with mass rejoicing in the Empire and England. You'd expect to see very close relations between England and Spain, and wouldn't anticipate an armada. This son undoubtedly receives the world's best education, has everything they could ever want, and is spoiled by his mother and father. Mary is probably delighted as well--she was quite fond of Edward. I imagine she makes a glittering marriage abroad, although not as glorious as she makes in OTL--dynastically, anyway--and Edward probably marries a pro-Hapsburg bride--Henry would want at least a princess for his precious boy, and no way is Queen Katherine okaying any French marriages. Perhaps Henry and Katherine suffer fights over the lack of a second son, but none of those are bad enough to result in annulment. She likely predeceases him, leaving him unwed in his 40s. As an acceptable, Catholic, grieving widower, he has his pick of Europe (unlike after Jane Seymour died) but he might choose to marry in the country due to infatuation. I doubt it's to Jane, though; he probably picks a Howard girl or one of the other court beauties. By this time he's already got his leg injury, but maybe he still has one or two children. He goes down in history as a failed general, wasteful in spending, a king governed mainly by his favorites, and his popularity only having been bolstered by personal charm and the blameless behavior of his first wife. Perhaps notations are made to his defense of the Church, his piety, and his scholarship, but he's virtually unknown as a political force, seen as more of a playboy/scholar-king type.England does not become a Protestant country, although it likely has possessions in the New World. Presumably Son inherits without controversy, and I can't begin to predict what would happen from there. Note also in this case the Scottish line diminishes in importance, and the Hanoverians likely never take the throne.
What english king broke away from the catholic church?
Henry VIII, so he could get divorced from his wife.