The 76-year-old made the comment during a Q&A session at Edgbaston Cricket Ground during the day-night Test against the West Indies on Saturday.
Speaking to VIP guests during a break in play, the BBC Test Match Special commentator claimed that knighthoods were handed out like "confetti" to notable West Indian players.
Boycott then told Sky Sports News presenter Gary Newbon, hosting the session: "Mine's been turned down twice. I'd better black me face."
The comment caused uproar among cricket fans, which prompted Boycott to write on Twitter: "Speaking at an informal gathering I was asked a question and I realise my answer was unacceptable.
"I meant no offence but what I said was clearly wrong and I apologise unreservedly. I have loved West Indian cricket my whole life and have the utmost respect for its players."
The BBC subsequently confirmed that Boycott will not be asked to resign from his pundit's role following the controversy.