Yes, an insurer can at the request of the policyholder do a named driver exclusion of a spouse from an auto insurance policy in Arizona. We verified this with the AZ Insurance Department, the insurance regulatory body for Arizona. Not all insurance companies have filed for this ability though.
Their representative stated that there is nothing in Arizona law that would preclude an insurer from doing this (in fact there are some circumstances when they must allow you to do so). However, some insurers might be reluctant to do so because there may be liability to the insured spouse.
Neither the AZ Insurance Department or CarInsurance.com can provide legal advice, so you might want discuss with an attorney if you are financially vulnerable by excluding your spouse, i.e., could you be sued if your spouse causes injuries or damage to someone, even if he is not driving your covered vehicle, if you would like to exclude your spouse from your car insurance policy.
The Insurance Department of Arizona also mentioned that as a resident of Arizona you should be made aware that most auto insurers already include an "Household," "Family," or "Intra-Family" Exclusion which limits Bodily Injury Liability coverage available to family members (or residents of the household) injured as a result of the negligence of another insured on the policy. Arizona law permits insurers to impose this Exclusion limiting such coverage to $15,000 per person/$30,000 per occurrence (Arizona's minimum financial responsibility limits), despite the amount of Liability coverage otherwise purchased under the policy.
Arizona Revised Statute (ARS) 28-4009(A) Subsection 3 gives information on your question regarding excluding someone from your car insurance policy. Here it states:
By agreement in writing between a named insured and the insurer the policy may exclude as insured a person or persons designated by name when operating a motor vehicle. The written agreement between the named insured and insurer is effective for each renewal of the policy by the insurer and remains in effect until the insurer agrees in writing to provide coverage for the person or persons who were previously excluded from coverage.
Another section of AZ law that speaks about excluding a named individual is ARS 20-1631 Subsection F if you would like to read that as well. This portion of law is about cancellation or failure to renew a policy and limitations of liability.
So by written agreement, the insurer and the insured may agree to exclude coverages if a particular person were to operate the insured motor vehicle. From case law in Arizona a named driver exclusion however cannot extend to UM/UIM coverage.
As a reminder, Arizona financial responsibility laws do require that if you drive a car in Arizona, you must have automobile insurance for the required minimum limits. The required coverages in Arizona are Bodily Injury and Property Damage Liability coverages. The minimum required limits are 15/30/10 which stands for $15,000 per person and $30,000 per accident for Bodily Injury and $10,000 per accident for Property Damage.
If your spouse is going to be excluded from your vehicle than he or she would not have any coverages if she/he were to operate your vehicle. The spouse thus needs to have insurance coverage on the vehicle(s) that he/she were going to drive. You can get an Arizona auto insurance quote here to see how much it would cost to insure your car.