Does backbiting, unheard by the person backbitten, and followed by seeking forgiveness, invalidate or diminish the fast, and does it necessitate making up for it by feeding the poor or fasting after Ramadan?

Backbiting is to mention about your Muslim brother something that he dislikes in his absence. And your statement to your brother, "These are manners," is understood as an aspersion on his character. Therefore, you must repent to Allah and seek forgiveness from your brother. Backbiting does not invalidate fasting according to the majority of scholars, so your fast is valid, and you are not required to offer expiation or compensation.