It's really state-dependent whether hospitals and/or groups can discriminate against DOs.
If you look at the rules and regulations in Texas, hospitals can not exclude DOs just because they are DOs
http://www.tsbme.state.tx.us/rules/mpa/chapter151.htm#151051
§151.051.Discrimination Based on Type of Academic Medical Degree or Certain Religious Tenets Prohibited
(a) A hospital, institution, or program that is licensed by the state, is operated by the state or a political subdivision of the state, or directly or indirectly receives state financial assistance may not differentiate in regard to a person licensed under this subtitle solely on the basis of the academic medical degree held by the person.
As for court cases
"Court decision in Audrain County, MO, established the right of DOs to practice as complete physicians and surgeons in a county hospital."
http://www.studentdoctor.net/do/timeline.asp
"... in 1959 the American Hospital Association (AHA) decided to change its longstanding policy barring joint-or-mixed-staff institution from membership. However, this reversal occurred only after the association became the focus of intense pressure from public hospitals that were being forced by court or legislative action to allow DOs access to their institutions and to give them staff appointments." - Chapter 9, page 143. Gevitz, Norman.
The DOs - Osteopathic Medicine in America 2nd ed. John Hopkins Press 2004