Who was the pope refused to annul Henry III marriage?
A it was Henry VIII and B it was Pope Clement VII who annulled the marriage between Henry and Catherine of Aragon.
Edit: It was Henry VIII, and Pope Clement VII, but... the Pope refused to annul the marriage between Catherine of Aragon and Henry VIII.
Originally Catherine had been betrothed to Arthur, Henry's older brother, but Arthur died before he became King. Henry VII (Arthur and Henry-to-be-VIII 's father) then paid the Pope (the one before Clement VII) a lot of money to say that Henry could marry Catherine, despite the Bible saying that it was illegal to marry your dead brother's wife. This rule was bent by Catherine swearing on the Bible, that she had not 'consummated' the marriage, which means she had not had intercourse with Arthur following the ceremony.
So Catherine married Henry, and he was crowned King Henry VIII.
However, the role of the King is to produce an heir (and a spare male in case the crown prince dies), and none of Catherine's boys survived! Only one child, a girl, survived- Princess Mary.
Meanwhile, Ann Boleyn sashays onto the scene- the long and the short of it is that Henry decides he wants her as his wife instead of Catherine, who is fast becoming too old for child bearing, and Ann Boleyn is young and pregnant.
So Henry writes to the Pope to ask him to annul the marriage between him and Catherine, saying that she had lied when she had said she had not had intercourse with Arthur, which is why only one of her children had survived.
Unfortunately for Henry, Catherine of Aragon's nephew, Charles V of Spain, the Holy Roman Emperor, had been warring with the Papal states- and had Pope Clement VII under house arrest! The pope was therefore unlikely to want to grant Henry this annulment!
A further reason for Clement's reluctance was that the Pope was meant to be infalliable- meaning, never wrong. If he said that Catherine had lied, and Henry's marriage to her should never have happened (therefore giving an annulment), the Pope would be admitting that the previous Pope had been wrong to grant the marriage. Which goes against their rules of infalliability.