A video posted on YouTube shows John Delaney in a pub singing "The Ballad of Joe McDonnell" about a member of the Provisional Irish Republican Army who died in a 1981 prison hunger strike.
"I do accept that if I have upset anybody here, I'm sorry," Delaney told Irish broadcaster RTE.
The video shows Delaney singing only the end of a song that earlier features inflammatory lyrics about Britain.
"When you sing a song like that, you don't believe in every word that is in the song," Delaney said. "I sing a large number of songs, maybe five or six different ones. It's normally done in a private way when there is a sing-song. It's a typically Irish thing we do. We sing songs amongst our group and you expect it to be kept to the group."
The Provisional IRA killed nearly 1,800 people during a failed 1970-1997 campaign to force Northern Ireland out of the United Kingdom, then renounced violence, and surrendered its weapons in 2005.
Although singing about the Irish republican prisoners, Delaney stressed: "I am not somebody who supports violence."
"My grandfather fought in the (Irish) Civil War and he also fought in the War of Independence," Delaney said. "I have always said I have a nationalist background."
Delaney complained about the sly nature of the recording, which appeared to be taken after Ireland beat the United States 4-1 a week ago in a friendly.
Delaney has become embroiled in a sectarianism row as Ireland prepares to host England in June for the first time since a 1995 game in Dublin was abandoned due to violence.
Asked about Delaney's singing, English Football Association general secretary Alex Horne said after a FIFA meeting in Belfast: "I have seen the apology, I haven't seen the video.
"From our perspective, we have a really close working relationship with our own fans and we are looking forward to the game in June. We don't want to see or hear any chanting like we witnessed in Scotland and we will be working very carefully ... to make sure that message is with the fans and everyone attending the game."
Sectarian taunts have long been heard at England matches, with anti-IRA chants by England fans at last week's away friendly against Scotland prompting an apology from coach Roy Hodgson.
Rob Harris can be followed at www.twitter.com/RobHarris