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.

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.

The pope's reasons for refusing to grant a divorce were largely political. Henry VIII prided himself on being a good Catholic. He was awarded the title Fidei defensor—defender of the faith—by Pope Leo X for his withering attacks on Martin Luther. Not unreasonably, then, Henry believed that he was in good standing with the Church and could therefore count on papal support for his intended divorce.

However, Clement VII was unable to offer much in the way of assistance to Henry, even had he really wanted to. At that time, the pope had much more serious problems to worry about, such as his bitter political dispute with the Holy Roman Emperor Charles V. Relations between the pope and the emperor had deteriorated so much that imperial forces invaded and sacked the city of Rome, causing widespread death and destruction. Clement was forced to hide in the Castel Sant' Angelo, where he was eventually imprisoned for six months.

Unsurprisingly, Clement had other things on his mind than a highly improper request from a monarch who wasn't even a particularly major player in European politics. Besides, Charles V was Catherine of Aragon's nephew, and he was deeply upset at Henry's shabby treatment of his aunt. Had Clement granted Henry a divorce, it would have antagonized the Emperor even more, and that's the last thing he needed at that time.