Blue Nile Inc. stockholders have approved the online jeweler's $500 million sale to Bain Capital Private Equity and Bow Street LLC. Votes were tallied Thursday and the "yes" votes won with 8.8 million votes to about 571,000 "no" votes, according to a filing with the Securities and Exchange Commission. The deal will make Blue Nile a private company again after 13 years on the public market. The acquisition remains subject to certain "customary closing conditions" and is expected to close in the…