Does a step-parent have the legal right to spank a step-child?

This is not a step-parent issue per se.  The real issue is, is corporal punishment illegal? 

While not has accepted as it once was, it is legal and not abuse if it is not violate the law prohibiting endangering children, or rises to the level of a criminal act. 

"Endangering children" is administering corporal punishment or other physical discipline, or physically restraining the child in a cruel manner or for a prolonged period, if the punishment or discipline is excessive under the circumstances and creates a substantial risk of serious physical harm to the child.
§ 2151.031. [Civil Code]

It is a criminal act to administer corporal punishment or other physical discipline, or to physically restrain the child in a cruel manner or for a prolonged period, if it is excessive under the circumstances and creates a substantial risk of serious physical harm to the child.  Further, it is a criminal act to administer unwarranted disciplinary measures to child if there is a substantial risk that if conduct is continued it will seriously impair the child's health or development.
§ 2919.22. [Criminal Code]

Based on your facts, I don't think this is a police/child services matter.  However, as a parent he may try to go court and have a judge grant in effect a "no spank zone" at home; that is prohibit corporal punishment in the home as a condition of continued custody.  Additionally, while this is probably not a criminal matter, it doesn't mean the he can't file a complaint with the authorities.  If so, they will be legally obligated to investigate even if no charges are ultimately filed.

At this point, a consultation with an experienced family law attoreny would be in order.