Is expiation (kaffarah) obligatory for not fulfilling a promise made to one's wife and affirmed by invoking Allah's name, after the wife agreed to waive it?

One should not make a covenant with Allah and His solemn pledge, fearing that they might not fulfill it. Rather, one should make a covenant with themselves, because if they breach their own covenant and word, it is less grave than breaching the covenant and solemn pledge of Allah. One must repent to Allah Almighty for not fulfilling a covenant and pledge.

If you intended an oath by your statement: "Upon me is the covenant and solemn pledge of Allah," or "I swear by the covenant and solemn pledge of Allah," then the preponderant opinion among scholars is that you owe an expiation for breaking an oath (kaffarat yamin (definition)). However, if you did not intend an oath, then you only need to repent, and no expiation is required of you. In this case, it is recommended to give charity.