Yes.
There are different classes of pain medications. Those that are addictive, such as narcotics, are regulated by the Drug Enforcement Agency (DEA), and require a specific license in order to be prescribed.
Anti-inflammatories such as ibuprofen and Naproxen do not require a DEA prescribing license. If a doctor is licensed to practice medicine, they are able to prescribe anti-inflammatories and other, non-addictive pain medications.
Most Physicians and Nurse Practitioners have a DEA license, so they are able to prescribe most types of regulated narcotics.
Some narcotics are strong enough that their use is usually monitored by a Pain Specialist, which may require a referral from your doctor.
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Currently, in the united states, the answer is no. Some states do allow chiropractors to prescribe over the counter medication, but no state allows chiropractors to prescribe legend drugs. New Mexico is one state that has attempted to allow limited prescriptive authority to chiropractors, and may in the future allow such prescriptive authority.