The singer/song writer has also asked radio stations to stop playing the song which was written about the conflict in Northern Ireland.
The song, taken from the 1979 album Armed Forces, is one of Costello's best-known hits and received endless plays across all BBC radio stations.
However, the line in question includes the lyric: "Only takes one itchy trigger/ One more widow, one less white n*****."
Costello examined that it was what his grandfather was called while serving in the British Army.
“It’s historically a fact," he said. “But people hear that word... and accuse me of something that I didn't intend."
Released in 1979, Oliver's Army was played unedited on radio stations for decades, however, many took the decision to bleep the lyrics as the word became increasingly taboo.
On his last tour, Costello rewrote the song to address being "cut down by the censors", singling out the BBC, which attracted criticism for editing the song in 2013.
"They're making it worse by bleeping it, for sure," he added. "Because they're highlighting it then. Just don't play the record.”
Costello added that radio stations will "do him a favour" by not playing the track again.
"Because when I fall under a bus, they'll play She, Good Year for the Roses and Oliver's Army," he said.
"I'll die, and they will celebrate my death with two songs I didn't write. What does that tell you?"
Costello’s cancellation comes after the Rolling Stones confirmed they would no longer play the song Brown Sugar due to controversy surrounding the song's lyrics.
The hit track was originally titled 'Black P****', but Mick Jagger decided before releasing it that the title was too 'nitty-gritty' and it was changed to Brown Sugar.
The 1969 song depicts scenes of slavery and sexual violence, including lyrics telling of a slave driver whipping a group of women, and has come under fire in recent years for its comments about slavery.
Keith Richards said the band made the decision to retire the track as they don't want to get involved in 'conflicts' about the lyrics, while Mick Jagger hinted that the song may return to their setlist in the future.
The last time the Stones played the hit track, which has sold 2,700,000 times in original sales since its release, was on August 30, 2019, at Hard Rock Stadium in Miami, Florida.