The suspension announced Wednesday is indefinite.
NASCAR senior vice president of racing operations Steve O'Donnell said Clements made an "intolerable and insensitive remark" during the course of an interview.
No other details were given, and Clements did not go in depth in a statement released Wednesday night.
"I apologize and regret what I said to the NASCAR writer and to NASCAR, my sponsors, my fans, and my team," he posted on his Facebook page. "NASCAR has a Code of Conduct that everyone must follow and I unintentionally violated that code. I will not get into specifics of what I said but my comment to the writer was in no way meant to be disrespectful or insensitive to anyone or to be detrimental to NASCAR or the NASCAR Nationwide Series. I will do what I need to do in order to atone for my error in judgment. "
Clements finished 33rd in Saturday's race at Daytona after dropping out early with an oil line problem.