The company offered "deep apologies for causing discomfort," saying it would take care to manage the Twitter account more carefully.
In English, the tweet sent Sunday conveyed "A very merry unbirthday to you!" from a song in the Disney film "Alice in Wonderland."
But in Japanese, the message read "congratulations on a not special day."
The tweet quickly drew sharp criticism from readers who questioned why Disney would offer such a message on the anniversary of the bombing, which killed more than 74,000 people on August 9, 1945.
Japan holds solemn ceremonies each year to mourn the victims.
Nagasaki was bombed three days after the first atomic bomb struck Hiroshima. Japan surrendered on August 15, ending World War II.