The first requirement for joining the EU is wanting to join. A far shot for the USA, but we assume this hypothesis in the question.

European Union has defined its [Conditions for Membership][1] in 1993. They are often refered to as the [Copenhagen criteria][2].

>Countries wishing to join need to have:
 
> -  stable institutions guaranteeing democracy, the rule of law, human rights and respect for and protection of minorities;
 
> -  a functioning market economy and the capacity to cope with competition and market forces in the EU;
   
> -  the ability to take on and implement effectively the obligations of membership, including adherence to the aims of political, economic and monetary union.

> The EU also needs to be able to integrate new members.

I would think that the **political criteria** would be met quite easily. There might be some discussion, and possibly needed adaptation, about the situation of Native Americans, about the political rights for American citizens living in Puerto Rico, Guam and other territories that are arguably excluded from presidential and Congress elections, or about racial policies (Europe doesn't recognize races and might debate if it can admit a country were Positive Action is a thing). But overall, I don't think these are obstacles that good faith on both sides wouldn't overcome.

The **economical criteria** are another story: given the US abyssal public debt, it would need to engage into efforts to reduce its public deficit. Subsidies for homeland agriculture and industries would have to be abandoned. Difficult, long-term and unpopular economic reforms to accomplish.

About the **geographical criteria**:

> Article 49 (formerly Article O) of the Treaty on European Union (TEU) or Maastricht Treaty states that any European country that respects the principles of the EU may apply to join. Countries' classification as European is "subject to political assessment" by the Commission and more importantly—the European Council. 

Basically, it is up to UE to decide what country is "European" or not. The answer has been "no" for Morocco, "yes" for Cyprus and Malta, and is still subject of heated debates for Turkey. Because there is [no precise definition][3] of the borders of Europe, I would think that ultimately it would be a political decision and if European are motivated to include the USA, they would make a case to allow them to join (based on history or values or whatnot). Same as they would allow Iceland to join without much geographical nitnitting. Anyway, territories likes the Azores or French Guyana are [already][4] part of the Union (considered as *European Outermost Regions*) while Greenland, Bermuda, Falkland Island or New Caledonian are *Oversees Countries and Territories* of the Union. 


  [1]: https://ec.europa.eu/neighbourhood-enlargement/policy/conditions-membership_en
  [2]: https://en.wikipedia.org/wiki/Copenhagen_criteria
  [3]: https://politics.stackexchange.com/a/11852/15897
  [4]: https://en.wikipedia.org/wiki/Special_member_state_territories_and_the_European_Union