But, still most of these certifications are based on exams and standards established by the states -- and then then states make agreements among themselves whether they will recognize each other's certifications.
Without looking, I don't know if California and New York share reciprocity for medical licensing or not -- but for the sake of argument, let's assume they DON'T.
A doctor moving from New York to California who wished to practice medicine there, would have to take the California State Medical Board Exam to be granted a license.
Nothing in the provision you cited would compel CA to recognize NY state board exams or licenses.
Seems like it would be a stretch to say that it would apply to HS diplomas.