Can a company claim compensation on an international level for breaking international law?

Can a company based in the EU claim economic compensation corresponding to the wrongly increased tariffs?

Usually, the country in breach of GATT obligations does not have to pay monetary compensation (damages). However, it is noted that the country in breach has to come in line with the rules in some other manner and stop misbehaving. The remedy stated in DSU article 19 is merely a recommendation to come in line with international obligations and no more than that. If the violation continues then the effect is retaliation from the Swedish side which can also lead to cross-retaliation from the US. DSU Article 19 calls for a recommendation “bring the [inconsistent] measure into conformity” with the legal obligation in question.”

The second sentence of Article 19 also authorizes Panels or the Appellate Body to “suggest” specific ways in which the offending member could implement the general recommendation. Punitive retaliation from the Swedish side would mean (as under GATT XXIII:1(b)) “compensatory” level of retaliation authorized for a legal violation is no greater than the level of retaliation authorized for non-violation nullification and impairment” leaving the Swedish company in a rather obscure and difficult position.

Although it is true that neither the EU and nor the US courts will uphold the international law in cases where national sovereignty is at stake we have come to know from precedence that EU courts are more likely to uphold international law.