Tldr; Henry claimed it wasn't a divorce, but that he was never actually married in the first place. Thus, divorce was definitely still bad, but not something he'd done, he was just guilty of sleeping with a woman who wasn't his wife, but who the Pope had made him think was his wife.

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A longer answer:

Essentially, the answer to this is that Henry VIII did not approve of "divorce", did not want a "divorce", and in fact never got "divorced". And so after his second marriage (and fifth marriage) ended, the Church at no point said divorce was okay.

(Today the Church is a bit less strict; they still see divorce as bad, but divorcees can remarry in a Church, as long as they didn't cheat with each other during their previous marriages. So while most divorcees can remarry in an Anglican church, even, say, the heir to the throne, son of the leader of the Anglican Church, who cheated on his wife and then, after his divorce and her death, married the woman he'd been having an affair with, would not be allowed to get married in an Anglican Cathedral. Not that I'm talking about anyone in particular here...)

What Henry wanted and argued for was an annulment; essentially, for the Church to say he had never been legally married in the first place. It may seem very similar to a divorce to our perspectives, but to Christians at the time (and indeed, today; the Catholic Church still gets petitioned for annulments and sometimes grants them, though it's not easy) divorce was wrong, whereas annulment was okay.

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Some background:

When Henry was young, his older brother Arthur died, imperilling the English-Aragonese alliance that Henry VII had made by marrying Arthur to Princess Catherine of Aragon.

But what was the solution? Marry Catherine to Arthur's younger brother, the new heir, Henry.

However, there was a problem. Catherine was Henry's sister-in-law, and the Catholic Church would not allow a man to marry his sister-in-law.

So both Henry VII and Isabella of Spain (Catherine's mother) asked the Pope to annul the marriage; essentially, saying that Arthur and Catherine had never, actually, legally, been married. Arthur and Catherine had only been married 20 weeks, and he'd been sick the whole time; Catherine and her governess claimed Catherine had never slept with Arthur, which may be true due to his illness, and so they'd never consummated the marriage. (Though they asked for dispensation from the Pope even if the two had slept together, just in case).

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Prince Henry didn't want to marry Catherine, but she was keen on marrying him. After his father's death, he suddenly changed his mind and they got married.

They were married for many years, mostly happily. But their son died as a baby (at 7 weeks), and most of Catherine's pregnancies were stillbirths. They only had one surviving daughter.

Once Catherine was 40 (Henry was 5 years younger than her), it was obvious she'd never give birth to another child. Henry's father had ended a long period of civil war due to issues over succession; Henry was obsessed with ensuring he had an heir. He had a daughter of course, Mary, but at this point no woman had ever held the throne (officially; Empress Matilda, only daughter of a previous King Henry, had claimed the throne only for most of England's nobles to ignore her and crown her cousin King, leading to a long war between them that was eventually ended with her agreeing that she wasn't Queen, but that her son would be her childless cousin's heir).

There were three options; get Mary to marry and have a son who could inherit from his grandfather (unreliable), legitimise Henry's bastard son (unpopular), or get a new wife.

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Henry had (apparently) become obsessed with a verse from the Book of Leviticus (20): “‘If a man marries his brother’s wife, it is an act of impurity; he has dishonored his brother. They will be childless." Henry believed (or claimed to believe, though it is quite likely he did) that the Pope should never have let him, as a teenager, marry his brother's wife, and that God had cursed their marriage to punish Henry for this sin.

Thus, if this marriage wasn't considered legitimate in the eyes of God, it wasn't a real marriage and should be annulled.

The woman Henry was in love with, Anne Boleyn (sister of a woman he'd previously had an affair with) was obviously very in favour of this, as were her father and his allies at court.

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So, Henry asked his ministers to convince the Pope to grant him an annulment on the basis that his marriage to Catherine was illegal. Not a divorce (which would imply the marriage was legal, but they didn't want to be married anymore). They failed. This created an opportunity for priests interested in reform and a break with Rome (what would later be called Protestants, but at this time were generally referred to as Reformers) to rise in power and influence over Henry, arguing that he should be the leader of the English Church, and that the Pope was denying what God wanted him to do. When the Archbishop of Canterbury died, Henry got a reformer, Thomas Cranmer, made the new Archbishop.

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Soon, Henry married Anne, and Parliament ruled on religious law, clarifying that Henry's annulment was definitely okay and legal. The loyal Catholic Lord Chancellor resigned, and was replaced by a pro-reform lawyer named Thomas Cromwell.

One important difference of Annulment vs Divorce was about Princess Mary. If Henry and Catherine's marriage had been valid, that would mean she was, well, a Princess. But as Henry's ministers had ruled that Catherine and Henry were never really married, that meant Mary was just a bastard, no more legitimate than Henry's bastard son. Any children by Anne would instead be his firstborn legitimate children. (As it happened, Anne later fell from grace, was accused of having sex with multiple men including her brother, and witchcraft, and was executed. Later Henry's government passed a law declaring he could just decide the order of succession however he liked, and that Anne's daughter Elizabeth was a bastard.)

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A short note on Henry's second "divorce"; again, this was actually an annulment. Henry and his fourth wife both claimed to have never slept together, and thus the marriage had never been consummated. So it could be annulled, allowing him to marry again.

Over time reformers like Cranmer and Cromwell pushed Henry further and further away from Catholic ideas. It went back and forth (and several reformers were executed on trumped-up charges by Henry later) creating the more Protestant Church inherited by his son, under whose rule reform sped up.