It is important to know that Henry would have parted ways with the Catholic church possibly anyhow, without the romance with Anne.
Cromwell was Henry's right hand man, bookkeeper, lawyer, "fixer", friend. He had discovered that the monasteries and abbeys of England were not paying their fair share to the Crown, Henry, as they were supposed to do. They owed rents for lands, which they had stopped paying. They were getting wealthy, and had piles of money, jewels, gold and silver bars, plates, goblets, which they were keeping to themselves and not sharing with the Crown.
Rumors were widespread of sexual activities at the monasteries and abbeys.
Henry was a pious and religious man when he was younger, and could not stand hearing of this.
When Cromwell suggested that the coffers of the monasteries and abbeys needed to be raided, Henry was only too glad to do it.
Reason one was the paragraph above.
Reason two was that Henry needed the money.
Henry VIII's father, Henry VII, had left him a wealthy man when he died. And had left England's coffers wealthy. But that was all gone now. Henry VIII had built numerous grand palaces all over England. Only one of them of them stands today, St James's Palace in London.
The grandest was probably Nonsuch Palace, next to that in magnificence was his Palace at Whitehall.
Henry VIII had a taste for expensive, luxurious living. Not so, his father. Henry VIII had bought tapestries, fine furnishings for his palaces; had thrown lavish parties. He had also outfitted his large army with armor, weapons, horses, and built many ships for his royal navy.
So now Henry needed money, badly. Raiding the monasteries and abbeys would allow him to take possession of all their money, treasures and property. So he did it.
Reason number three: Henry needed a male heir, plus was enraged when the Pope would not do what Henry asked him to do in order to achieve this.
In the process of leaving the Catholic church, Henry made himself the wealthiest monarch in all of Europe.
He had supreme rule over England without the Pope telling him what he could and could not do.
He divorced wives whenever he wanted and took a new one.
Yes, I believe Henry would have parted ways with the Catholic church and the Pope even without Anne.
Was she a small part of the reason? Yes.