What if instead of having only a daughter, Catherine of Aragon gives birth also to a sterile son, who succeeds him but cannot have his own children, what would Henry VIII do in this scenario..
Katherine died of cancer, though. I'm not convinced she'd live much past 1536.
I agree, though, Henry would probably remarry, whether he'd be able to produce another son, though post-1536/1537, I'm not sure, because in OTL Edward VI was born in 1537 and Henry didn't have children with any of the remaining three of his wives.
Back then, it was ALWAYS the woman's fault. The gender, the lack of children, etc. What a son means is no annulment. when (if) Katherine of Aragon kicks it in 1536 (without all the stress, probably later), he'll remarry some pretty (maybe Christina of Denmark will look more favorably on him?) foreign beauty and stay with the church.