Queen Elizabeth says that she can't interfere with 2 people wedded in the eyes of God or something, but her father changed the religion of the entire country in order to make divorce legal so he could marry her mother.
As you say, Henry VIII did virtually invent a religion to make divorce legal, but he didn't do so arbitrarily; basically, he had people research old records and eventually found a document stating that the king had complete power over England, which he interpreted to mean that the king was free to establish a new religion. But both Henry VIII and Elizabeth I knew that they had to stay in the parameters of Anglicanism; there were horrifically bloody disputes between the Anglican and Catholic Churches, and Elizabeth probably figured that repeatedly claiming authority to continually change the rules of a comparatively young belief system would result in uprisings and possible revolt. By respecting the sacrament of marriage, she kept herself safe.
Henry did not amend church doctrine to allow divorce as we know it. The dissolutions of his marriages, while they could be seen as de facto divorces, were officially annulments or pronouncements that the marriages had not been valid in the first place. Until very recently, the Church of England forbade divorcees to remarry until their ex-spouse died.