Why did the Catholic Church not allow Henry VIII to get a divorce from Catherine of Aragon?
Henry did not want a divorce, but an annulment. The real reason Henry VIII could not get a papal annulment of his marriage was the political and military power of Catherine's nephew, Emperor Charles V. It was actually a bit more complicated than just wanting the dispensation, there were many complications involving the marriage itself, the balance of power in Europe and Catherine's family. First of all, Catherine of Aragon had been wife to Henry's older brother, Arthur, the heir to the throne of Britain. Arthur died four months later, and a treaty was signed between Henry VII and Catherine's parents (Ferdinand and Isabella of Columbus fame) allowing for a marriage between Catherine and the new heir to the throne Prince Henry (who of course became Henry VIII). This marriage of a man to his brother's wife required a papal dispensation. It was not a necessity, since Catherine swore her marriage to Arthur had never been consummated, but the monarchs requested and received dispensation as a formality This became important later during the annulment attempt.
The marriage was beset with political difficulties before it even took place, and afterwards produced no male heir to Henry VIII's throne. A miscarriage was followed by a son who died very shortly, followed by a second miscarriage and a second son who died soon. Mary was born next, followed by apparently two more miscarried pregnancies. Henry wanted an heir (and had a keen eye for the ladies, anyway), and by the time his romance with Anne Boleyn became known Catherine was 42 and unable to bear children. So Henry decided to petition the Pope for an annulment on the grounds of childlessness and her first marriage to his brother. Of course, there had been three children, one of whom was still alive, but politically this didn't matter to Henry, he wanted a male heir. As to the first marriage, a dispensation had been granted so it didn't matter, and Catherine swore all her life the marriage had not been consummated. Given Prince Arthur's health issues that seems likely.
Henry's problem was that Catherine appealed to her nephew Charles of the House of Hapsburg. He was son of Catherine's sister Joanna and Philip of Burgundy, and also happened to be King of Spain, Holy Roman Emperor and ruler of Burgundy, Naples, Milan, etc. Essentially, Charles V was the major power on the mainland of Europe, and he let the pope know that if he granted an annulment to Henry VIII he, Charles V, would sack Rome. The pope spent six years prevaricating and wrangling over details with Henry until, in 1533, Anne Boleyn became pregnant. Henry VIII simply ordered the Archbishop of Canterbury, Thomas Cranmer, to grant the annulment and broke with the Roman Church.
Divorce was not allowed. Henry fell in love with Anne Boleyn and this is one of the main reasons why he wanted an annulment. When Henry fell in love with Anne, Catherine was no longer able to bear him any more children. She had several miscarriages but had one healthy daughter. Not a son, but a daughter. Henry wanted a male heir and Catherine could not give him this heir. He petitioned the Pope, not for a divorce, but for an annulment. This is because he believed (or perhaps it was an excuse) that marrying his brothers wife was not seen right in the eyes of God. Catherine swore she did not consummate the marriage to Henry's brother, therefore she was never really married to him. After 6 years of being denied the annulment by the Pope, Henry had Thomas Cranmer, the archbishop of Canterbury, grant the annulment.
This sort of depends upon your own attitudes towards the Catholic Church and the values it stood for back in those days.
The official reason that the Church did not allow Henry to divorce is that divorce was absolutely forbidden in those days. The Church simply did not allow them.
But lots of monarchs got divorces in similar circumstances --- they could petition the Pope to annul their marriages. So why not Henry? Historians who are more critical of the Church say that the Pope denied Henry because the Pope was related to the King of Spain, who was Henry's enemy. The claim is that the Pope wanted to make sure Henry would have no heir so the King of Spain would have a better chance of taking England over.