Henry VIII did not ask the Pope for a divorce; he asked for an annulment. The difference being that divorces are the termination of a legitimate marriage, while an annulment meant that the marriage was never legitimate in the first place.

Henry initially tried to have the annulment processed and approved by Cardinal Thomas Wolsey, who, it was argued, could issue annulments in England. The other problem Henry faced was that in order to marry Katherine of Aragon, he had sought and received a dispensation from the Pope. A papal dispensation was a document that allowed things normally illegal or against church law to happen without consequence - examples include allowing bastards to become priests, or for clergy to hold multiple benefices. In Henry's case, the dispensation was sought because there was historical and biblical precedent that stated a person could not marry their brother's wife. Henry's older brother Arthur had been married to Katherine before his death, and so to avoid any future legal/dynastic problems, Henry received the dispensation to 'violate' this law.

Fast forward 20 years, and Henry felt that his childless marriage was indicative that regardless of the dispensation, he (Henry) had violated God's law, which no Pope could dispense, and therefore the marriage had to be annulled. The Pope disagreed, and further, the Pope was currently the prisoner of Katherine's nephew, Charles V. So even if the Pope wanted to annul the marriage on the grounds that he had no power to dispense with God's law, he couldn't because he was under the thumb of the Holy Roman Emperor. I elaborated a bit on some of the above arguments here.