The chairwoman of the Republican National Committee said Tuesday that the party wouldn’t return donations from Steve Wynn, its former finance chairman who faces allegations of sexual misconduct dating back decades, unless an investigation by his company’s outside directors finds wrongdoing.

In an interview with Fox News, Ronna Romney McDaniel called the allegations against Mr. Wynn, published by The Wall Street Journal last week, “deeply troubling” and noted that Mr. Wynn had stepped down from his post as finance chairman...