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Could a state fail to recognize a divorce --- for example imagine that a state believes that one can only get divorced for specific reasons. Would they have to recognize a divorce in another state that where none of the three reasons were met --- I would suspect that they would have to recognize it even if they found it morally reprehensible.

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Excellent question. Without doing a lot of research, I will say that it would likely come out as follows: If a state permits divorce (for reasons X, Y, and Z), then the concept of divorce is not repugnant to the public policy of that state. Therefore, if would have to give FF&C to a divorce on other grounds from another state.

Now, if a state were to abolish divorce entirely, then it might well come out differently.