It would be more nearly correct to say that the judge does not have the authority to accept or reject a plea of not guilty by reason of insanity. That is a defense, and as such is a matter to be determined by the trier of fact, namely the jury. The sole--and unusual--exception would be if the defendant opted for a bench trial by the judge himself. In that event, the Judge would, as the trier of fact, decide if the defendant was, in fact, not guilty by reason of insanity.
The above answer states that in many states, the defendant must prove that he or she is insane. This is not true. In fact, in EVERY state an insanity defense is an affirmative defense that the defendant must prove by clear and convincing evidence. The prosecution has no burden in ANY state to prove a defendant is sane; it must only argue that the defendant has not met the burden of proof of an insanity defense, and of course prove that the defendant is guilty of the crime beyond a reasonable doubt. Please note that the burden of proof for the prosecution is greater than that for the defendant; still, as an affirmative defense, the burden of proof of an insanity defense falls on the defendant, not the prosecution.
The only rules that differ from state to state are the definitions of insanity. In my state, the common law M'Naughten Rule is still in effect; which states that at the time of the offence, the defendant was incapable of distinguishing between legal and moral right and legal and moral wrong. Other states have other definitions, but in every state, it is necessary for the defendant to prove that, but for his mental incapacity, the crime would never have been committed. The fact that one suffers from a mental defect is not in and of itself sufficient to absolve one of responsibility.
No, a judge does not have to accept this plea. The exact rules surrounding such pleas vary from state to state in the United States, but a judge does not have to accept that the defendant is in fact insane.
In many states, when a defendant pleads not guilty by reason of insanity, the defendant must prove that he or she is insane. The burden of proof rests on the defendant. In other states, the prosecution must prove the defendant is sane beyond a reasonable doubt. Either way, the final decision is with the jury in the case of a jury trial. It is the jury that must determine if the defendant is sane.
Depending on how a case is being tried, either a jury or the Court (a judge) determines whether the defense plea will be accepted or rejected. The states evidence considerable differences in their statutes. For example, Idaho, Kansas, Montana, and Utah do not permit an insanity defense, but have retained the "gulity, but insame" defense. The American Law Institute (ALI) proposed a Model Standard in 1962 which is less restrictive than the M’Naughten rule, as it permits two relevant but distinct considerations: (1) Mental illness or defect that caused the defendant to not appreciate the wrongfulness of his act (Section 4.02 on impaired capacity), or (2) caused the defendant to not conform his conduct to what the law requires (Section 4.01, on excluding responsibility). The volitional test and the appreciation of wrongfulness are completely separate tests. The standard holds that a person may not be able to control their behavior while at the same time appreciating the criminality of their act.
The Mental Health America (MHA) organization supports the insanity defense (guilty by reason of insanity) and opposes the “guilty but insane” laws that do not allow the use of the insanity defense. Given the increasing number of mentally ill people who are incarcerated this makes sense, particularly in light of the MHA's position that mentally ill people should be diverted from jails and prisons and confined in suitable places for appropriate treatment.
An interesting film on the issue of competency to stand trial is "Nuts" starring Barbra Streisand.
He does not have the authority to do so.