Singapore reacted furiously in February when the refurbished frigate was named "KRI Usman Harun", lodging a diplomatic complaint with Jakarta and banning the vessel from its ports and naval bases.
Tensions escalated last month after the Indonesian navy dressed two marines as the executed bombers at a defence exhibition in Jakarta.
"Once again I apologise. We have no ill intent whatsoever to stir emotions. Not at all," military chief General Moeldoko, who like many Indonesians goes by one name, said in an interview with Singapore broadcaster Channel NewsAsia that was aired on Tuesday.
Moeldoko, however, said that the ship will not be renamed.
Singapore's Defence Minister Ng Eng Hen welcomed the apology on Wednesday, saying it was a "constructive gesture to improve bilateral defence ties".
This will "strengthen the mutual understanding and friendship that has been built up over many decades," he said in a statement.
Usman Haji Mohamed Ali and Harun Said were executed in Singapore for their roles in the March 1965 blast at a downtown office complex which killed three people and injured 33.
Indonesia considers the two men to be national heroes. Their attack was part of an effort by then Indonesian president Sukarno to stage an armed confrontation against the newly formed federation of Malaysia, which included Singapore.
In his Channel NewsAsia interview, Moeldoko said: "Indonesia didn't think that 'Usman Harun' would eventually turn into a polemic such as this."
"It is my responsibility as the commander-in-chief of the (Indonesian armed forces) to offer a clarification and to take steps to ensure that the situation does not escalate," he added.
Indonesia is Singapore's third largest trading partner, with total trade between the Southeast Asian neighbours reaching Sg$79.4 billion ($62.6 billion) in 2012.
Relations hit a low point in the late 1990s after the fall of former dictator Suharto, and his successor B.J. Habibie famously referred to the tiny city-state as a "little red dot" on the map.
Bilateral ties have improved considerably in recent years.