What is the ruling regarding a man's statement to his wife, "You are unlawful to me, and it is unlawful for me to approach you," especially if he denies saying it and swears to that effect, while his wife confirms his statement made in a state of anger?

The prohibition of one's wife refers back to the husband's intention. If he intended divorce, it becomes a divorce. If he intended ẓihār (a pre-Islamic form of divorce by comparing one's wife to an unmarriageable relative), it becomes ẓihār. If he intended an oath by Allah Almighty or did not intend anything, he is obligated to offer an expiation for an oath. The husband's statement, "It is forbidden for me to approach you," would likely indicate an intention of ẓihār or divorce if he had an intention. However, if he uttered the prohibition of his wife during extreme anger, to the extent that he was unaware of what he was saying, then nothing is binding upon him because accountability is lifted from him.