Could Henry go for a powerplay? Deny his brother a divorce in hopes that no son will be produced, allowing Pope Henry (or Adrian or whatever) to ascend the throne as King of England? That would be epic but probably very implausible.
No, because it would be about as subtle a tactic as a brick to the head, and all of Europe would see the obvious motivation behind his action and take steps to force him to back down (after all, this is the era of the King of Spain who invaded Rome and put the Pope under house arrest, though for more legitimate reasons).
Besides, remember that Popes were generally close to the end of their lives when elected. Arthur would not only be too old for a divorce by the time Henry might be elected to the Papacy, he would most probably be dead already with grandchildren surviving to keep on the family line.
But the real crux of the matter is that he would never have been elected. This era saw the office of Pope essentially used as a pawn between the Spanish and French kingdoms, with the country in the ascendancy managing to put a national of theirs on the Papal throne. On the handful of occasions when a Pope wasn't blatantly pro-Spanish or pro-French, the Conclave are not going to elect the heir to the English throne as Pope because he will be so flagrantly politically-charged that it would be a sincere threat to both kingdoms. A neutral Pope would always come from Italy, because of their heritage of packing the Cardinals' College, because of their strong and ambitious powerplaying families, and because of their ultimately minor status in European affairs, meaning any Italian Pope is never going to be able to successfully derail French/Spanish ambitions by promoting Italy, at least not in the way that a pro-English candidate not just might but surely would. Allowing Henry to be elected would be like the French and Spanish agreeing to shoot each other in the foot. For that very reason, Henry would likely never even be raised to a Cardinal's position - it prevents the very possibility of English interference in their affairs.