The Eleventh Amendment to the US Constitution has been interpreted (by the Supreme Court) as conferring a significant degree of sovereign immunity on the states. US courts will generally not award money damages or injunctive relief against US states unless they have waived immunity by state law. By itself, this already takes most of the teeth out of GDPR, since its remedies are basically all fines. While the European court system might refuse to recognize such immunity, and purport to issue judgments against individual states, such judgments would have no practical effect unless the US court system were willing to recognize the fines as valid.
Under Ex parte Young, the Supreme Court held that state officials may be enjoined from enforcing unconstitutional state laws, under the legal fiction that such enforcement is not an official act when the underlying policy is unconstitutional. One might therefore imagine trying to obtain injunctive relief indirectly. I doubt this would be successful. The US has no federal equivalent to GDPR and has not signed any sort of treaty or agreement with the EU regarding its applicability to the states. As a result, the Supremacy Clause is inapplicable, and there's no obvious line in the Constitution saying that states are required to respect extraterritorial foreign laws.
Such enforcement also raises significant separation of powers concerns. The US President enjoys more or less plenary power over diplomacy and foreign relations, with the sole exception of treaty ratification. If a US court were to apply a foreign law to a US state, this would interfere with the ability of the executive to negotiate a treaty to that effect. As a result, the state would likely argue that applicability of foreign laws is a political question to be resolved by treaty or federal law, and not by the court system.