There are a myriad of different laws allowing for prosecution for such damage to federal property. This include the general provision under 18 U.S.C. §1361 for any damage exceeding $1000.
1361. Government property or contracts
Whoever willfully injures or commits any depredation against any property of the United States, or of any department or agency thereof, or any property which has been or is being manufactured or constructed for the United States, or any department or agency thereof, or attempts to commit any of the foregoing offenses, shall be punished as follows:
If the damage or attempted damage to such property exceeds the sum of $1,000, by a fine under this title or imprisonment for not more than ten years, or both; if the damage or attempted damage to such property does not exceed the sum of $1,000, by a fine under this title or by imprisonment for not more than one year, or both.
There is also Veterans’ Memorial Preservation and Recognition Act of 2003 (18 U.S.C. 1369) which states:
(a) Whoever, in a circumstance described in subsection (b), willfully injures or destroys, or attempts to injure or destroy, any structure, plaque, statue, or other monument on public property commemorating the service of any person or persons in the armed forces of the United States shall be fined under this title, imprisoned not more than 10 years, or both.
(b) A circumstance described in this subsection is that—
(1) in committing the offense described in subsection (a), the defendant travels or causes another to travel in interstate or foreign commerce, or uses the mail or an instrumentality of interstate or foreign commerce; or
(2) the structure, plaque, statue, or other monument described in subsection (a) is located on property owned by, or under the jurisdiction of, the Federal Government.
(Added Pub. L. 108–29, § 2(a), May 29, 2003, 117 Stat. 772.)
Thus, there are ample laws that apply to such alleged crimes and the executive order merely makes enforcement a priority policy of the Administration. What is notable about this release is the reference to the crime. Note the second paragraph where the Violent Crime Task Force is cited as pursuing individuals for general destruction of public property. That would refer to not the Veterans’ Memorial Preservation and Recognition Act of 2003. I am not aware of litigation under the latter law but the task force may be avoiding any questions over whether the Jackson statue is “commemorating the service of any person or persons in the armed forces of the United States.”
There is an old legend that equestrian statues followed a code where one hoof is raised meant that the the rider was wounded in battle while two raised meant that the rider died in battle. (All four hooves on the ground meant the rider was unharmed). That appears to be an urban legend as shown by the Jackson statue with two hooves raised (Jackson died at 78 in bed surrounded by his family for what was called chronic dropsy and heart failure). The statue therefore could be questioned as being a celebration of Jackson’s military service, though he is wearing a uniform and was widely known for his military record. Indeed, the 1853 statue is the oldest full-scale equestrian sculpture in the United States and artist Clark Mills reportedly was inspired to celebrate in Jackson’s triumph over the British in the Battle of New Orleans in the War of 1812.
The availability of alternative grounds for prosecution seems to be preferred over triggering such nuanced interpretative challenges.
There is also a provision under D.C. law citing the $1000 damage threshold under Section 22-303:
“Whoever maliciously injures or breaks or destroys, or attempts to injure or break or destroy, by fire or otherwise, any public or private property, whether real or personal, not his or her own, of the value of $1,000 or more, shall be fined not more than the amount set forth in § 22-3571.01 or shall be imprisoned for not more than 10 years, or both, and if the property has some value shall be fined not more than the amount set forth in § 22-3571.01 or imprisoned for not more than 180 days, or both.”
However, the D.C. police chief has stated that he made the “tactical decision” not to intervene as some statues were destroyed or defaced.
It seems therefore that Section 1361 will be the main focus of the enforcement effort under the executive order.