No, and there is no reason for him to. Times have changed.
Once upon a time, 80 years ago, the Church of England totally disapproved of divorce. Meanwhile, "the Establisment" was not happy with Edward VIII and even the late George V said "the boy will ruin himself within twelve months". Oh boy, was he right! Edward wasn't interested in the duties of being king, left secret papers lying around to the consternation of officials, and liked Hitler. Nor did "the Establishment" like what it saw as his gold-digging twice-divorced girlfriend, Wallis Simpson. So the likes of the Prime Minister, Stanley Baldwin, and the Archbishop of Canterbury, Cosmo Lang, intrigued for months to get him to go.
They must have thought Christmas had come early when Edward made it clear that he wanted to marry Wallis. No way would it be right for the Supreme Governor of the Church of England to break its own rules by marrying a divorcee! There is no law about this, and there never has been, but it just "wouldn't be cricket". OK, the Church of England was basically started by Henry VIII so he could have a new wife, but ah, that was different - he was never divorced, he had marriages annulled, which isn't the same thing.
So at this point "the Establishment" could tell the newspapers and it could be a simple ultimatum. You can have the throne or you can marry Wallis, not both. To everyone's relief, Edward chose Wallis. There was also the little legal problem that according to the Marriage Act 1836, royals had to marry in church, and didn't have the option of a register office. So marrying Wallis and staying king was just impossible - they had to marry in church, and the Church wouldn't marry anyone who's already had a divorce.
Now wind on to 2005. The Church of England had changed its mind and will now remarry the divorced, and the Marriage Act 1836 was finally repealed in 1953. Of course Charles and Camilla checked up on all this before getting married to make sure Charles could still be king when the time comes. The Archbishop of Canterbury, Rowan Williams, would have been perfectly happy to marry them in church, and the reason they didn't was the Queen, who is very traditionalist about divorce and wouldn't have come to the wedding. So Charles and Camilla did the traditional C of E thing, getting married in a register office and having a church service of blessing afterwards.