According to
snopes.com the answer is no.
Washington is home to more equestrian statues than any other city in the nation, and it's significant that perhaps only 10 out of 30 or more follow the convention. Here's a quick look-see at various equestrian statues in Washington and how they fit or don't fit this theory. First, some statues that follow the "rule":
FRANCIS ASBURY: 16th and Mount Pleasant NW (1924). All hooves on ground; died in peace.
FIELD MARSHAL SIR JOHN DILL: Arlington National Cemetery (1950). All hooves on ground; died of leukemia.
GEN. ULYSSES S. GRANT: Union Square, at the east end of the Mall (1922). All hooves on ground; died in peace.
MAJ. GEN. WINFIELD SCOTT HANCOCK: Seventh and Pennsylvania NW (1896). One hoof raised; wounded in battle.
MAJ. GEN. JOHN A. LOGAN: Logan Circle, Vermont Avenue, 13th and P Streets NW (1901). One hoof raised; died in peace, twice wounded.
LT. GEN. WINFIELD SCOTT: Scott Circle, 16th and Massachusetts and Rhode Island NW (1874). All hooves on ground; died in peace.
GEN. PHILIP H. SHERIDAN: Sheridan Circle, 23rd and Massachusetts NW (1908). All hooves on ground; died in peace.
GEN. WILLIAM TECUMSEH SHERMAN: 15th and Pennsylvania and Treasury Place NW (1903). All hooves on ground; died in peace, pneumonia.
MAJ. GEN. GEORGE H. THOMAS: Thomas Circle, 14th and Massachusetts NW (1879). All hooves on ground; died in peace.
JOHN WESLEY: Wesley Theological Seminary (1961). All hooves on ground; died in peace.
And now some that don't:
GEN. SIMON BOLIVAR: 18th at C and Virginia NW (1959). One hoof raised; died in peace of tuberculosis.
MAJ. GEN. NATHANIEL GREENE: Stanton Square, Maryland and Massachusetts NE (1877). One hoof raised; died in peace, unwounded.
MAJ. GEN. ANDREW JACKSON: Lafayette Park (1853). Two hooves raised; died in peace.
LT. GEN. THOMAS J. (STONEWALL) JACKSON: Manassas (1940). All hooves on ground; wounded by own men and died.
MAJ. GEN. PHILIP KEARNY: Arlington National Cemetery (1914). One hoof raised; died in battle.
MAJ. GEN. GEORGE B. McCLELLAN: Connecticut Avenue and Columbia Road NW (1907). One hoof raised; died in peace, unwounded.
BRIG. GEN. JAMES B. McPHERSON: McPherson Square, 15th between K and I streets NW (1876). One hoof raised; shot and killed in battle.
BRIG. GEN. COUNT CASIMIR PULASKI: 13th and Pennsylvania NW (1910). One hoof raised; died in battle.
LT. GEN. GEORGE WASHINGTON: Washington Circle, at 23rd and K and Pennsylvania and New Hampshire NW (1860). One hoof raised; died in peace of cynache trachealis. Washington Cathedral (1959). One hoof raised.