He died on Monday, aged 89, in a nursing home in Newmarket-on-Fergus in County Clare, following an illness.
Bishop Casey resigned as Bishop of Galway following the revelation he had been making payments to a woman in the United States for a long period.
He had a relationship with an American divorcee and was the father of her son.
Peter Murphy was born in 1974 and was the result of Bishop Casey's affair with Annie Murphy.
Speaking on BBC's Evening Extra programme, Michael Kelly of The Irish Catholic newspaper described the event as "the beginning of the end of the extreme dominance of the Catholic Church in Irish public life".
When the relationship became public in 1992, Bishop Casey resigned his post and went to America.
He later moved to Mexico to learn Spanish, worked as a missionary priest in Ecuador and then worked in a parish in England before retiring to Galway 14 years later.
In 2006 he came home to live in Shanaglish, near Gort, County Galway, and then Carrigoran Nursing Home in County Clare in 2011.
Born in County Kerry on 24 April 1927, Bishop Casey was educated in Limerick and in Saint Patrick's College, Maynooth, from where he was ordained to the priesthood in 1951.
For the next nine years he worked as a curate in two Limerick city parishes before being appointed to the Irish Emigrant Chaplaincy Service in England.
He became well-known while working in London for sourcing adequate housing for thousands of Irish emigrants.
On Monday, the Catholic Church said "his passionate advocacy for social justice and for those marginalised by inequality gave him a significant public profile and his work in this area continued throughout his life".
He was appointed Bishop of Kerry in 1969 before taking on the larger and more high-profile diocese of Galway and Kilmacduagh in 1976.
"Perhaps the highlight of his time in Galway was his organising and hosting the visit of Pope John Paul II to the city in 1979 to meet and pray with 300,000 young people from all over Ireland," the Catholic Church said.
Bishop Casey was present at the funeral of his murdered friend Archbishop Oscar Romero of El Salvador when the liturgy was interrupted by bombs and intense gunfire which left almost 50 mourners dead.
He also voiced strong objections to the visit of US President Ronald Regan to Galway in 1984 because of "American foreign policy".
Archbishop Eamon Martin said:  "It is with great sadness that I learned today of the death of Bishop Eamonn Casey.
"I wish to sincerely acknowledge the contribution of Bishop Casey to the work of the Irish Catholic Bishops' Conference over twenty-three years during his time as Bishop of Kerry and as Bishop of Galway".
Former Bishop of Galway, Eamon Casey, who was at the centre of a scandal that rocked the Catholic Church in the early 1990s, has died.