His resignation after 21 years has been accepted by Pope Benedict.
Bishop Clifford, who was born near Dundalk in Co Louth and ordained as a priest in 1967, has worked closely with Cardinal Sean Brady.
However he is now stepping aside after receiving medical advice.
"It is with great sadness that I come to today's decision," he said.
Dr Clifford has held a variety of roles within the church, including as executive secretary of the Irish Catholic Bishops' Conference and as president of the church's pro-life crisis pregnancy counselling service Cura.
He has also undertaken work towards establishing closer relations with the Church of Ireland.
"Having grown up locally in the diocese - born in the parish of Lordship & Ballymascanlon in 1941 - and as priest and auxiliary bishop in the Archdiocese of Armagh, I have received great warmth and affection throughout my ministry from the people of the diocese," he said.
"I thank them for their support in my 21 years as auxiliary bishop."
He said Cardinal Brady had been a model colleague, adding: "His humanity, deep faith and natural humility bear witness to the most fundamental of Gospel values."