Father Fleming sidestepped the vow of celibacy when the former Anglican priest converted to Catholicism in 1987, one of only a handful of priests whose bishops have received papal dispensation permitting their ordination.
Now, in an unprecedented move, the National Council of Priests, representing about half the Australian Catholic clergy, wants the Vatican to reconsider its ban on married clergy.
Citing dwindling priest numbers, the council argued in its December submission to Rome that the right held worldwide by thousands of married clerical converts like Father Fleming ought to be extended to other married men.
Yet, Father Fleming, president of the soon-to-be-opened Campion College in western Sydney, feels there are equally strong arguments for maintaining the tradition of celibacy in the priesthood.
He does not necessarily subscribe to the view that the experience of married life makes for more empathetic priests.
"There is no greener grass on the other side of the fence," he said. "The advantages of celibate priests are many and the National Council of Priests is not suggesting it be done away with.
"Those positives include a witness to chastity in a fairly sexually obsessed society, greater portability and it's cheaper to upkeep a celibate male than a family.
"There are huge implications for the life of the church, not the least of which is who is going to pay for married priests. In Mexico some priests are so poor they live under the altar of the church. These practicalities shouldn't be ignored."
Eastern rite churches, whose structure and worship is similar to that of the Orthodox faith, permit married clergy.
The apostle Peter, the Catholic Church's first Pope, was married.
According to Paul Collins, a writer and former Catholic priest, the sacred vow of celibacy has been more about discipline than church doctrine.
"I just frankly think this citadel of celibacy is crumbling in front of us," he said.
"I'm quite prepared to bet that within the next papacy or the one after that, we will have married men as Catholic priests. It's not about priests being happy and comfortable; it's about serving the growing number of Catholics who want to celebrate Mass effectively."
The chairman of the Church's Bishop's Committee for Clergy and Religious, Bishop David Walker, said the National Council of Priests' statements did capture a feeling among some priests.
"I wonder if this is the proper time to make such a decision when there is so much unenlightened comments relating to celibacy and the abuse of children by priests," he said.
"While many professionals recognise there is no direct relationship, the attention given to the abuse phenomenon has created a false atmosphere around the issue of the celibacy of clergy," he said.
"I don't believe there will be any immediate change in the church's practice, but in terms of the pastoral care of the clergy I think the church should be constantly reviewing how it can assist priests in the celibacy which is demanded of them."