Nancy was sacked after just eight games in charge, prompting the Scottish club to bring back former boss Martin O'Neill until the end of the season.
Given, who made more than 600 senior appearances as a goalkeeper at several clubs, said on Saturday that O'Neill should have been given some more time before they brought Nancy on board ahead of a crucial week.
"Nancy took over that week and had an absolute holocaust in that week and it was just a nightmare from then on," Given said on the broadcast.
BBC presenter Jason Mohammad apologised for the remark later on in the broadcast, before Given apologised with a statement on social media.
"On live television this afternoon, I used a word that I didn’t fully understand the meaning of, and certainly won’t ever use again," Given wrote.
"We all have areas of ignorance in our knowledge, and I hope to use this as an opportunity to become better educated going forward.
"I am genuinely mortified and apologise unreservedly to everyone offended, and will be donating my fee from today’s show to the Holocaust Education Trust."