If there is a real business need and it is necessary for everyday, ongoing business, employers can require their employees to provide specific medical information. But if it's not, an employer can't require an employee to provide specific medical information.
An illustration comes in a recent $2 million settlement of a class action Americans With Disabilities Act (ADA) suit by former and present employees of Dillards' department stores. Dillards' required employees to provide specific medical information regarding mental illness, cancer and gynecological information in order to have their sick leave approved. If the employee refused to provide the specific details of their medical treatment, they were fired. This resulted in a $2 million class action settlement for present and former Dillards' employees.
You can read more about the Dillards' case: Requirement that Employees Explain Sick Leave Results in $2 Million Settlement.