Cooper shares fell 9 per cent in extended trading as the action was the latest indication that the merger was in trouble. The stock is now trading more than $9 off Apollo's $35-per-share offer price.
The deal has faced opposition from workers at Cooper's joint venture in China and from demands made by a U.S. labor union.
Cooper said on Friday that Apollo was looking to delay settling issues with its workers, belonging to the United Steel Workers (USW) union, who have asked that new agreements be drawn up between them and Apollo post merger.
Earlier this week, Cooper Tire shareholders approved its sale to Apollo.
"With their approval, we have met our conditions for closing," Cooper Chief Executive Roy Armes said in a statement.
"The complaint filed today is a necessary step in the process to assure that the terms of the merger agreement are met as required and that we do everything possible to get the transaction closed promptly," Armes said.