While, technically, the constitutional crisis would have been caused by him marrying a divorcee and being the head of a state religion that didn't approve of remarriage with living ex-spouses, the circumstances were likely important in motivating a hard stance on the policy: it involved the sort of situation that an apologist might have given as an example of why remarriage should not be allowed. Even current Church of England rules would not allow the marriage.
It is interesting that the story is often simply portrayed as him wanting to marry a American divorcee, likely leading to the sense in many readers unfamiliar with the circumstances that he wanted to marry someone who simply had had prior marriages, quite possibly with ex-husbands who were still in the US.