NPS CWS&SS does not show a Robert Forsythe with North Carolina troops but does show:

Robert Forsythe
Regiment Name 1st Kentucky Infantry
Side Confederate
Company C
Soldier's Rank_In Private
Soldier's Rank_Out Private
Film Number M377 roll 5

1st Regiment, Kentucky Infantry
1st Infantry Regiment was organized during the summer of 1861 for a one year term of enlistment. In August the 1st (Duncan's) Kentucky Battalion merged into the regiment, and in December totalled 800 effectives. Ordered to Virginia it fought at Dranesville under J.E.B. Stuart, then in March, 1862, was assigned to provost duty at Orange Court House. In April, with 496 men, it was placed in D.R. Jones' Brigade, Army of Northern Virginia. However, its term of service soon ended and the unit disbanded. The field officers were Colonel Thomas H. Taylor, Lieutenant Colonels Edward Crossland and William P. Johnston, and Major Benjamin Anderson.

Alexander Forsythe
Regiment Name 1st Kentucky Infantry
Side Confederate
Company
Soldier's Rank_In Surgeon
Soldier's Rank_Out Surgeon
Film Number M377 roll 5

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U.S. Civil War Soldiers, 1861-1865
Name: Robert Forsythe
Side: Confederate
Regiment State/Origin: Kentucky
Regiment Name: 1 Kentucky Infantry
Regiment Name Expanded: 1st Regiment, Kentucky Infantry
Company: C
Rank In: Private
Rank In Expanded: Private
Rank Out: Private
Rank Out Expanded: Private
Film Number: M377 roll 5

American Civil War Soldiers
Name: Robert Forsythe ,
Enlistment Date: 23 Apr 1861
Enlistment Place: Louisville, Kentucky
Side Served: Confederacy
State Served: Kentucky
Service Record: Enlisted as a Private on 23 April 1861.
Enlisted in Company C, 1st Infantry Regiment Kentucky on 23 Apr 1861.
Mustered Out Company C, 1st Infantry Regiment Kentucky on 14 May 1862 at Richmond, VA.

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1860 United States Federal Census
about Robert Forsythe
Name: Robert Forsythe
Age in 1860: 43
Birth Year: abt 1817
Birthplace: North Carolina
Home in 1860: Ledge of Rock, Granville, North Carolina
Gender: Male
Post Office: Knap of Reeds
Household Members: Name Age
Robert Forsythe 43
Mary Forsythe 38
Helon Forsythe 10
James Forsythe 7
Logan Forsythe 5
Robert Forsythe 3
William Forsythe 1