The country that did not establish a colony or colonies on the Caribbean islands during the 1600 and 1700 is Italy.
Spanish colonizers settled in America from the late 15th century until the 19th century. The Spanish colonies in the Caribbean were called the Spanish West Indies. These were mainly Cuba, Dominican Republic and Puerto Rico. Moreover, the term 'Spanish Caribbean' is used to distinguish it from the Anglophone Caribbean and the French Caribbean. The British colonizers invaded islands such as Nevis, Antigua, Anguilla and Bermuda. On the other hand, the French colonized Guadeloupe, Martinique, St. Martin and St. Barts, among others. Therefore, Spain, France and England established colonies on the Caribbean Islands during the 1600 and 1700 but Italy did not establish colonies there.