Does the Superior Court of Justice have inherent jurisdiction to order a party to undergo an assessment by someone who is not a “health practitioner”, as defined in s. 105 of the Courts of Justice Act?

Yes. 

Ziebenhaus v. Bahlieda, 2015 ONCA 471:

[9]          The Divisional Court concluded that s. 105 does not "occupy the field". It noted that the health sciences and patient care have evolved to include a wide range of assessments by experts who are not "health practitioners". Such assessments cannot all be characterized as diagnostic aids to the opinion of a "health practitioner". Precluding their use in the litigation context would be contrary to good public policy. In the light of these circumstances, the court said, at para. 45: 

The only conclusion that can be drawn from these circumstances is that section 105 does not completely "occupy the field" in the sense that it makes no provision for physical and mental examinations that are routinely used in the care and treatment of injured persons, and in litigation, that are conducted by persons who do not qualify as "health practitioners" under section 105. Accordingly, there is a gap in the statutory provisions regarding the entitlement of a party defending an action to require a plaintiff to submit to such examinations.

[10]       On the issue of whether an order for an assessment by an individual who is not a "health practitioner" would be contrary to the intent of s. 105, the Divisional Court determined that it would not. 

[11]       I see no error in the Divisional Court's analysis and conclusion.

[12]       As set out by the Supreme Court of Canada in R. v. Rose, [1998] 3 S.C.R. 262, at para. 133: 

[T]he inherent jurisdiction of superior courts is a significant and effective basis for preventing abuse of the court's process and ensuring fairness in the trial process. This enduring and important jurisdiction of the court, if it is to be removed can only be accomplished by clear and precise statutory language.