The episode title refers to not only the Latin definition, but also to 18 U.S.C. § 1385. "Use of Army and Air Force as posse comitatus. Whoever, except in cases and under circumstances expressly authorized by the Constitution or Act of Congress, willfully uses any part of the Army or the Air Force as a posse comitatus or otherwise to execute the laws shall be fined under this title or imprisoned not more than two years, or both."
The idea of doing a production of several of Shakespeare's so-called History Plays and calling it "The War of the Roses" originated with New York's The Public Theater, which in 1971 mounted a combined, all-night, marathon version of the Henries and Richard III as a fundraiser for the then-struggling theater. Martin Sheen, who plays President Bartlet on The West Wing (1999), was a Public Theater repertory member during the early days of the company. Sheen had some of his earliest acting successes while at the Public, including well-reviewed performances as Romeo and Hamlet.
The song used at the end of the episode was originally written for the Royal Shakespeare Company production of "The Life and Adventures of Nicholas Nickleby" - the staging at the end of the play in the episode was heavily influenced by the RSC's production. The star of that production, Roger Rees (who also played Nicholas Nickleby in the subsequent television adaptation of the play) was a frequent guest star on The West Wing (1999), as British adviser and later British Ambassador Lord John Marbury.
Definition: Posse Comitatus (Late Latin): posse: "to be able (to have)" comitatus: "a company" or "armed retinue". The power of the county, or the citizens who may be summoned by the sheriff to assist the authorities in suppressing a riot, or executing any legal precept which is forcibly opposed.[Websters]