St. Augustine, FL? 1565?
Nombre de Dios, Panama? 1540?
Santo Domingo, PR? 1496?
If you simply google this question google brings up an automated answer of St. Augustine, but other sources claim the other two answers I listed. I don't want to know the longest continual European settlement. (which I think St Augustine is?) I want to know the first European settlement widely accepted by historians as a year round and permanent (at their time) settlement with the ability to self sustain.
Wouldn't it be L'Anse aux Meadows in Newfoundland? Or Brattahlíð if you count Greenland as part of North America?
Well like /u/canaman18 said Santo Domingo was founded in 1498 and fits your description of year round and permanent. Now that being said it is worth noting that there were Norse settlements in Greenland and even, for a short time, Canada by about 980AD that were year round an permeant. The last settlement died out in about 1430AD.
Source: http://www.wikipedia.org/wiki/Norse_colonization_of_the_Americas
Santo Domingo in the Dominican Republic was founded in 1498 I believe the Santo Domingo you refer to is the one in the Dominican. In Continental North America I know Veracruz was founded in 1519.