"I ask the forgiveness of all those to whom I may have been unfair and to those who I may have caused heartache," Mixa wrote in the letter.
Roman Catholic apology
His offer of resignation also came as Roman Catholic leaders in Britain apologised for child abuse by clergy, saying the scandal has brought shame on the church.
Archbishop of Westminster Vincent Nichols, the head of the church in England and Wales, said on Thursday that the crimes of some priests were a "profound scandal".
"They bring deep shame to the whole church. But shame is not enough. The abuse of children is a grievous sin against God," he said.
Nichols made the apology in a statement issued on behalf of bishops, to be sent to parishes across the country.
The Vatican has been rocked by a worldwide clerical sex abuse scandal in which the church hierarchy is accused of covering up for paedophile priests for decades.
Nichols' statement was issued as Pope Benedict XVI officially accepted the resignation of an Irish bishop over the cover-up of clerical abuse of children.
James Moriarty apologised for not challenging the culture of secrecy in the church.
"Again I accept that from the time I became an auxiliary bishop, I should have challenged the prevailing culture. Once more I apologise to all survivors and their families," James Moriarty said in a statement.
"The truth is that the long struggle of survivors to be heard and respected by church authorities has revealed a culture within the Church that many would simply describe as unchristian," he added.
Moriarty was auxiliary bishop in Dublin from 1991 to 2002.
He tendered his resignation as bishop of Kildare and Leighlin to the pope in December after a devastating report revealed decades of abuse by clergy in the Dublin archdiocese and a cover-up by Catholic authorities there.
The pope has also accepted the resignations of bishops Brendan Comiskey, Donal Murray and John Magee.