Quote:
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Originally Posted by Merriam-Webster
APOLOGY usually applies to an expression of regret for a mistake or wrong with implied admission of guilt or fault and with or without reference to palliating circumstances <said by way of apology that he would have met them if he could>.
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An apology is supposed to be about remorse and regret, right? If you don't feel it, why on Earth would you lie about it? What's the point of apology if it is insincere?
Being disingenuous to others about your feelings is, in my opinion, just horrible, horrible, horrible. You're not only being dishonest with others, but you're being dishonest with yourself, which is much, much worse, and there's really no reason for it.
Don't base an apology on numbers of those offended, don't do it just for the sake of smoothing things over, and, above all, don't do it if you're not sorry. To apologize for any reason other than remorse is insulting....for all involved parties.
Edit: I got off on a tangent there and realized I forgot to answer the question. If it's a misunderstanding than probably not. I would try to clarify and fix the misunderstanding. After that, if they're still offended (and you're not sorry 
), then well...that's their problem and not yours.