Is what the husband said to himself, "If the wife menstruates and does not become pregnant, I will give her such and such an amount," considered a vow that he must fulfill? And is it permissible for him to spend from this amount on household needs after the wife refused to take it and then returned it to him to save it for her?

What you uttered is not a vow, but merely an internal monologue and an intention. Therefore, nothing is incumbent upon you, and you are not obligated to pay the amount to your wife. You may reclaim it from her if you paid it assuming it was obligatory, as she is not entitled to it. However, if you paid it as a donation and she accepted it, and then returned it to you as a trust (wadi'ah), then you must safeguard the trust and not dispose of it except with her permission, because it has become her property.