If an exempt employee has been out sick with a physician note for several days and our company doesn't offer sick time at all, aren't we still required to pay him or does he have to use his vacation time.

You are required to pay him. However, you may also use his vacation time if you elect. That's the legal answer.

Having said that, if that is the intent of your vacation benefit, you should just be calling it Paid Time Off. Not providing sick pay, then being able to charge the employee with vacation time because you are legally required to pay him, seems very unfair to me. If the company chooses not to provide sick pay benefits, then the company should be willing to reap what they sow and pay the ill or injured employee what they owe under the law, without the employee having to use vacation time. OK, off soapbox. :p