Ohio Rev. Code Ann. § 339.02
(C) In making appointments to a board of county hospital trustees, both of the following apply with respect to the individuals who may be appointed:
(2) A physician may serve as a member, including a physician who is authorized to admit and treat patients at the hospital, except as follows:
(E) With respect to the initial appointment of members to a board of county hospital trustees, all of the following apply:
(F) Except as provided in division (G)(2) of this section, all of the following apply with respect to vacancies on a board of county hospital trustees:
(G)