Is expiation binding upon someone who made a mistake while certain that he did not swear an oath not to commit this mistake, yet his soul suggests to him that he did swear such an oath?

What is incumbent upon you is to repent to Allah, seek forgiveness, and supplicate for the one whom you backbit or wronged, not to inform him. This is because the correct view is that there is no need to inform him; rather, seeking forgiveness and mentioning his good qualities in the places where you backbit him are sufficient. You are not required to offer an expiation for an oath (kaffarat yameen) or anything else, because doubt does not necessitate an oath or a vow. Perhaps the doubt you are experiencing is from the whispers of Satan, so do not pay attention to this doubt or even think about it.