Catherine of Aragon was the aunt of Charles V, King of Spain and Holy Roman Emperor. Prior to Henry VIII’s application, Charles V had captured Rome and effectively imprisoned Pope Clement VII (such respect for the Vicar of Christ and Prince of Apostles!). The Pope could therefore do nothing without the approval of Charles V.
The succession of Mary Tudor would mean Charles V’s own first cousin would rule England. If her succession were challenged internally (which is what happened to Matilda, who was the last princess to attempt succession), Charles V could internvene militarily. He would prop Mary up on the throne and even induce her to marry a Spanish prince, thus turning England over to the ruling family of Spain. That is why the succession of Mary had to be prevented at all costs.
The Pope actually didn’t deny Henry’s application. He refused to act upon it, and the case dragged out for years. At one point the Pope floated the idea that perhaps Henry could be allowed to commit polygamy to solve the problem. It came to the point that England could wait no more, and so the Archbishop of Canterbury decided the case and issued the annulment. Parliament passed the Act of Restraint in Appeals to end the Pope’s appellate jurisdiction over all legal matters in England- thereby preventing Catherine from appealing the annulment to Rome.
Clement VII was the nephew of Leo X, one of the worst popes in history (He’s the guy who mirrored the money changers that Jesus drove from the temple because he was the pope selling indulgences when Martin Luther came to Rome) Clement was, like Leo, a DiMedici (a powerful Italian family like our Kennedys) and believed he could get away with practically anything. the DiMedicis had cardinals killed so they could get their family popes elected; they also were much like the Kennedys in impregnating young girls and having them murdered too.