What is the ruling of Islamic law on one who swore an oath upon the Qur'an to his wife that he has nothing with another person, while intending something other than what his wife intended, for the purpose of preserving his family, despite feeling doubt about his oath?

If your purpose is to preserve the family, then this is a valid purpose that permits lying to the wife. And if you need to swear an oath, then it is better to use equivocation rather than outright lying. Nevertheless, if you swear, there is no sin upon you. And if lying and swearing upon it is permissible for this purpose, then it is even more permissible to report something whose accuracy is doubtful and to swear upon it if needed.