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.