No, for two reasons.
Firstly, Article I, Section 10(3) of the US Constitution prevents any US state from entering any international organisation without the permission of Congress:
No State shall, without the Consent of Congress, lay any Duty of Tonnage, keep Troops, or Ships of War in time of Peace, enter into any Agreement or Compact with another State, or with a foreign Power, or engage in War, unless actually invaded, or in such imminent Danger as will not admit of delay.
Secondly, even if the state concerned secedes from the US, Article 49 of the Consolidated Treaty on European Union will throw up an issue of geography:
Any European State which respects the values referred to in Article 2 and is committed to promoting them may apply to become a member of the Union. The European Parliament and national Parliaments shall be notified of this application. The applicant State shall address its application to the Council, which shall act unanimously after consulting the Commission and after receiving the consent of the European Parliament, which shall act by a majority of its component members. The conditions of eligibility agreed upon by the European Council shall be taken into account.