There is a strong contender for the father of your Israel Eastwood, b. 1818, SC. In the 1820 Orangeburg, SC census (indexed as Earwood, but actually, Easwood) a John Eastwood (Easwood) has a son under 10 and 2 young daughters, and probable wife in his household. Unfortunately, the records for this county are extremely rare for this time frame. However, on the SC State Archives page there are some documents online:
http://www.archivesindex.sc.gov/
There are some search results for a John Eastwood and a J. Eastwood in Orangeburg in 1819 and 1797 (do a search for Eastwood and all the given names come up).