The 41-year-old Fazilova had been called up as a late replacement for injured Sugrakhanim Mugabilzada, but three days before the archery competition was due to get underway, she decided it was better not to compete.
"Everyone knows that she is good," her husband, Natiq Fazilov told reporters.
"She came 17th in the European Championship in 2012, but still rumours are going around. We decided that it would be better if she did not take part.
"We don't like being the centre of a scandal. Of course my wife is disappointed but to stop the gossip we did not have any other choice."
She will be replaced by 25-year-old Yaylagul Ramazanova.
The inaugural European Games, which kicked off on Friday, run until June 28.
(Reporting by Karolos Grohmann; editing by Toby Davis)
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