I find a birth record on Ancestry.com for a James Abernathy to Robert Abernathy and Christine Tillman in Virginia in 1746.Not sure if that is the same one who married Elizabeth Cox or some other James Abernathy or if this is even a reliable source:
Name: James Abernathy Father: Robert Mother: Christine Tillman Birth Date: 1746 State: VA Country: USA
Edmund West, comp.. Family Data Collection - Births [database on-line]. Provo, UT, USA: The Generations Network, Inc., 2001.
Description: The Family Data Collection - Births database was created while gathering genealogical data for use in the study of human genetics and disease