That is of course a bunch of Assad propaganda. But even them were more accurate at times:
Syria’s foreign ministry said on Sunday that an American oil company had signed an agreement with Kurdish-led rebels who control northeastern oilfields in what it described as an illegal deal aimed at “stealing” Syria’s crude. [...]
Most of money probably went to the Kurds, but while it is true that US company was involved, the US rejects the claim that it stole anything:
“Syrian oil is for the Syrian people and we remain committed to the unity and territorial integrity of Syria,” the [US] spokesperson added. “The United States government does not own, control, or manage the oil resources in Syria. The populations in areas liberated from ISIS make their own decisions on local governance.”
Of course, Trump in his Tweeter-level diplomacy phrased that as “keeping the oil,” but of course the US hardly needs that extra oil, compared to the Kurds.
... and some it (with no US help) went to ISIS which also controlled a bunch of fields for a while.
Assad could of course go after (all of) them for "restitutio ad integrum", which he can certainly do against the domestic groups, but of course he chooses not to do that against the Kurds at the moment, preferring to blame (just) the US.
This is not too unlike what was arranged in Iraq for the Kurds there, except then Turkey was involved in the (pipeline) exports. And yeah Iraq too is or at least was complaining (in 2014 at least).
Iraq’s Oil Ministry expressed “deep regret and astonishment” on Friday over plans announced by the autonomous Kurdish region to begin piping oil to Turkey without central government approval. [...]
It also rebuked the Turkish government for allowing the Iraqi-Turkish pipeline system to be used to pump and store crude oil produced in Kurdistan without Baghdad’s approval. [...]
Iraq’s constitution mandates that all Iraqi oil revenues go through the central government in Baghdad, and the Kurds then receive 17 percent, although they frequently complain that they have received less than that. [...]
The ministry on Friday threatened legal action against any companies that trade in “smuggled” oil or gas from Kurdistan without going through [Iraq’s State Organisation for the Marketing of Oil].
But they didn't go as far as blaming the Turkish gov't of theft. But the showdown continues to this day it seems.
By the way there's not much evidence that the Trump-era company managed to do much if anything concrete in Syria, other than serve the Assad regime an easy piece of propaganda. Most of the concrete oil deals done by the Kurds in Syria have been either selling back to Assad or in a smaller part selling to the KRG in Iraq, which resells it.
As for the legal Q, Syria could in theory sue the US at the ICJ, if the latter consented to be sued there. The legal question has exactly the same answer for similar claims, e.g like when Ukraine says Russia stole some of their grain (and sold it to Syria among other destinations). (The practical difference is however how many US troops are in Syria, vs how many Russian troops are in Ukraine, in terms of how credible the claims of direct theft are. Of course, there are almost certainly some and possibly many Ukrainians in Russian occupied territories interested in selling their produce to someone, rather than having it rot.)
