Williamson had said that the Nazi's had not used gas chambers to kill Jews held in concentration camps during the second world war and that 300,000, rather than the widely believed six million, Jews were killed in the Holocaust.
"To all souls that took honest scandal from what I said, before God I apologise," he wrote.
"As the Holy Father has said, every act of unjust violence against one man hurts all mankind."
Response to demands
The bishop said that he was responding to a demand by the pope to reconsider his comments made on Swedish television in January.
The remarks shocked many people in Argentina, where Williamson headed a seminary, and he was asked to leave by the government.
Argentina has one of the largest Jewish populations outside of Israel.
The letter, written from London, was said to be released by the Ecclesia Dei commission, established in 1988 to persuade traditionalist Catholics to return to the Vatican.
Williamson belongs to the Society of St Pius X, a traditionalist sect, and was excommunicated after he was ordained in an unauthorised ceremony 20 years ago.
Pope Benedict XVI had lifted Williamson's excommunication, along with that of three other bishops, in January to bridge to divide with traditionalists that occurred after liberal reforms in the 1960s.
The pope later said that the Holocaust denial was "intolerable", in an attempt to appease Jewish outrage.
The Vatican said that Williamson's comments were not known to the pope when he lifted the excommunication.