Updated; with nearest misses now Aruba and Grenada. Some explanation follows: (1) There are four countries that comprise the Kingdom of the Netherlands: Sint Maarten, Aruba, Curaçao and the country of the Netherlands. The country of the Netherlands in turn consists of the part in mainland Europe, and also three Caribbean islands of Bonaire, Sint Eustatius, and Saba. (2) The French Caribbean islands are the departments of Martinique and Guadeloupe, and the collectivities of St Martin and St Barthélemy, which all have the same status as regions of metropolitan (ie European) France and so are effectively not territories but a part of France proper. Hence the closest island is listed under the heading "France".
fair enough but martinique and guadeloupe are not territories of france, but rather departments, equivalent in status to alaska or hawaii, and they are definitely less than 800km away from the u.s. virgin islands.
Sint Maarten has been a country since 2010 when the nation of Netherlands Antilles, of which it had ben a part, dissolved. See the 2nd comment from the top, posted Nov 9 2014, which gives more detail.
Russia is only 2 miles (3 kilometers) away from the US because of the islands of Big Diomede (owned by Russia) and Little Diomede (owned by the US) which lie in between the two countries in the Bering Strait.