Short answer: I think you can do this.
Long answer:
1630.14(c):
(c) Examination of employees. A covered entity may require a medical
examination (and/or inquiry) of an employee that is job-related and
consistent with business necessity. A covered entity may make inquiries
into the ability of an employee to perform job-related functions.
(1) Information obtained under paragraph (c) of this section
regarding the medical condition or history of any employee shall be
collected and maintained on separate forms and in separate medical files
and be treated as a confidential medical record, except that:
(i) Supervisors and managers may be informed regarding necessary
restrictions on the work or duties of the employee and necessary
accommodations;
(ii) First aid and safety personnel may be informed, when
appropriate, if the disability might require emergency treatment; and
(iii) Government officials investigating compliance with this part
shall be provided relevant information on request.
(2) Information obtained under paragraph (c) of this section
regarding the medical condition or history of any employee shall not be
used for any purpose inconsistent with this part.
It sounds to me like you can make this inquiry. However, you should keep in mind that, although we normally associate the ADA with qualified individuals with disabilties, even unqualified individuals with no disability are covered by the provisions regarding tests. Therefore, an improperly requested medical examination of a mentally and physically sound person can still lead to trouble.
The crux of the matter is that your test request must be "job related" and "consistent" with "business necessity". An exam is generally consistent with business necessity if the exam seeks to diagnose a problem that is seriously effecting business efficiency, is needed to ensure the safety of co-workers or the general public, or is useful in showing that an employee needs (is qualified for) an accomodation. Given that we are talking about a police patrolman with a car crashing problem, I think you can ask that employee to get a medical examination because the employee works in a position in which his or her medical condition could lead to a direct threat to the safety and well being of other employees or the general public. You can request a health examination when an employee's performance has declined over time to rule in or out that the cause of that performance decline is health related, and that situation also applies. If you have a workers' comp employer friendly medical facility around that advertises and is versed in ADA issues, I'd send this person to them.