Public Law 810 refers to Part II, Chapter 23 of U.S. Code 38 which says that the government should, when requested, pay to put up monuments or headstones for unmarked graves for three groups of people:
(1) Any individual buried in a national cemetery or in a post cemetery.
(2) Any individual eligible for burial in a national cemetery (but not buried there), except for those persons or classes of persons enumerated in section 2402(a)(4), (5), and (6) of this title.
(3) Soldiers of the Union and Confederate Armies of the Civil War.
Public Law 85-425 was passed 23 May 1958, entitling the widows of deceased Confederate soldiers to military pensions:
To increase the monthly rates of pension payable to widows and former widows of deceased veterans of the Spanish-American War, Civil War, Indian War, and Mexican War, and provide pensions to widows of veterans who served in the military or naval forces of the Confederate States of America during the Civil War.
No where in either piece of legislation are Confederate soldiers declared to be American veterans.
Since they are grouped together in the same sentence, Union soldiers cannot be American veterans either.