Would Antonín Dvořák be considered a 'classical composer', specifically thinking about the very American "New World Symphony".
I have never studied music, so not sure of the different categories. E.g.: 'classical', 'romantic'.
The term Classical in terms of music has two main distinct meanings that have been mentioned in the comments. In the broadest sense Classical refers to music that is related mostly to the history of Western music composition. This is mainly a popular definition which can be seen in how music stores organize music to sell.
The other main definition of Classical refers to a period in Western music history that very roughly corresponds to the 18th century. The main figure associated with the Classical period would be Mozart.
Using these two definitions I would say that Dvorak is a classical composer and the New World symphony is a classical piece, in the first, broad sense of the word, but neither the composer or the music is of the Classical period.
As mentioned already, this would be categorized as late Romantic music. The Romantic period follows the Classical periods during is associated with a large group of composers that begins with Beethoven.
The Oxford Dictionary of Music (2nd ed. 1994), a one-volume reference, offers these definitions for Classical:
Obviously Dvořáks is missing the first but matching all other criteria, and his symphony „From the new world“ is not exceptional compared to his other works (except regarding popularity). Note that (1) is the narrow term, (4) the wide definition probably causing your confusion and (3) fitting phrases like classical Jazz. I admit, that I can't get much out of (2).