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According to the Daily Mail, a spokesperson said: "The British Council expects the highest standards of our staff and in accordance with our code of conduct we have started disciplinary procedures with the individual concerned".
Her comment was liked by 27 people but was condemned by many.
One woman wrote: "You are a total idiot Angela, to look at the picture of a child and be filled with nothing but hatred for them. You are a disgrace to humanity," according to The Sun, which has access to the Facebook page where Gibbin's posts appeared.
Gibbins, who is understood to earn £80,000 a year in her role, reportedly replied: "I'm sound in my socialist, atheist and republican opinions. I don't believe the royal family have any place in a modern democracy, least of all when they live on public money. That's privilege and it needs to end."