Greene County was not in existence until it was set off from Washington County, PA as of 9 Feb 1796.Anyone who before then lived in what is now Greene, would have been living in Washington County, which was created 28 Mar 1781 from Westmoreland.
So anyone who lived during the Revolutionary War in what became Greene County, was in Westmoreland County for the first part of the war (1776-1781), and in Washington County during the last couple of years (1781-1783).
That narrows down what County Militia your ancestor may have been counted in, except that Virginia's Monongalia County also recruited from the same area since VA claimed that part of present-day PA.
So your man's official record could have been made as being in the militia of Westmoreland or Washington County, PA or, Monongalia County, (West) Virginia.
You mention a plaque placed by DAR.You should, then, be able to find your man listed as a DAR patriot ancestor on the DAR's website, which you can search free: http://www.dar.org/library/online_research.cfmhttp://www.dar.org/library/online_research.cfm
--read and click through the introductory pages to get to the search page.Be sure to check under both PA and VA. A source just might be given, once you find the right man.
For PA entries, you can check the Penna. Archives' ARIAS card file of Revolutionary War soldiers that includes militiamen and State Troops and many soldiers who served in the Continental Line: http://www.digitalarchives.state.pa.us/http://www.digitalarchives.state.pa.us/
--click the link at bottom, and on the next page scroll down to the link to Revolutionary War soldiers listings, which are alphabetical.
If you find your man in the PA listings, look at the bottom of the card, which might give you a reference to Series, Volume and Page of the ~published~ _Pennsylvania Archives_, most likely in Series 3, 5, or 6.You can then find the Volume and page on the Internet Archive site, free, as well as other places, such as www.fold3.com using a free trial period offer.The latter site is also searchable by name, but "Smith" would just be an awful lot of looking-at-results unless his first name was very unusual.