Fr Skinner, who has waived his right to anonymity, told the BBC he was 27 and a curate at St Clare's in Newton Aycliffe when Gibson, the church's priest, indecently assaulted him.
He was also approached by a woman who said she saw Gibson kiss a Vietnamese man who was new in the area and seeking support.
Fr Skinner, who now works for the Scottish Episcopal Church, reported both incidents to John Habgood, the then Bishop of Durham, in 1981 or 1982, but after Gibson denied any wrongdoing the Bishop told the young curate to "refrain from listening to parish gossip" and stop "causing problems".
Fr Skinner said he refused to work with Gibson but when he went for a job in another parish, the "first thing" the interviewer asked him was why he had reported Gibson.
"I was blackballed," he said. "They made it so I could not do my job."
In her report, Dr Hill said Fr Skinner had described "a process of insidious demolition of his professional reputation and personal character", culminating in him relinquishing his orders for a time, meaning he could no longer work as a priest.
Fr Skinner said his church file was "doctored" to make him him "look like a bad boy".
His file included letters from Gibson "saying I had done this or that" but no record of the complaints Fr Skinner had made about Gibson.
Dr Hill said it was "of particular concern" there was no record of Fr Skinner's meetings with Bishop Habgood, who died in 2019 having become the Archbishop of York from 1983 to 1995.
She said it was "more likely than not" that meetings did take place but the bishop "failed to act appropriately", adding that the abuse suffered by another man at the time "may have been affected" if Gibson had been "more robustly challenged".
"The lack of record either indicates an inadequate response at the time, based on a presumed failure to recognise the seriousness of the situation and/or disbelief of it, or possible misguided decisions to later remove allegations and concerns from personal files," Dr Hill added.
She said it was one of five "red flag moments" to catch Gibson which were missed by church officials.
Fr Skinner said the church was "vicious with a civil veneer" in how it "covers up" abuse.
He said he was not "seeking revenge" but rather wanted the church "held to account" so it can "be better".
The current Bishop of Durham, the Right Reverend Paul Butler, said he would also like to know why the diocese acted the way it did in the 1980s.
The bishop said: "We do not know why the records of these meetings were not made and kept, and we cannot account for the decisions made by others.
"What we can say is that were such conversations to take place today, careful records of those meetings would be made, and would be kept."
'Culture of cover-up'
He said he could see the "harm and damage" caused and "it is clear" Fr Skinner was a "whistleblower who was not properly listened to".
He added: "I apologise once again for the failures of the church in that respect.
"[Fr Skinner's] sense of being treated unjustly is understandable.
"However, I disagree that a culture of cover-up still exists in the way that it did then. If further relevant information were available, even if that portrayed the practice of the Diocese of Durham at the time in a very negative light, we would share it.
"Today we professionally and appropriately deal with any complaints made, and in this Diocese, we report to the police everything that we should."