There is some evidence to support this statement, but a lot depends on what you mean by "closer" and by "classical English".

English, like all languages, is constantly changing. Accents, Dialects, Grammar, Vocabulary, Spellings, change with time and vary by location. There have always been a large number of accents and varieties of English in England. (So it is hard to define such a thing as "Classical English".)

At one time, when colonists arrived in America from England, they spoke the same varieties (numerous varieties) of English as the people in the parts of England from whence they came.

Over time, the English language kept changing, both in America and in England, and in neither country is the language the same today as it was when the English colonists arrived in America.

It is probably true, however, that in the case of accents, American accents have changed less since colonial times than English accents.

This is for several reasons:

The denser population of England seems to have allowed for more rapid shifts in language than the less dense population of America;

The large concentration of a dense population in London (with much of the massive growth of London coming after American colonization) has had a considerable impact on the development of English in England (and has probably impacted to some degree many of the English accents and dialects well outside of London). Most of the American colonists came from areas of England other than London, so the increasing influence of London accents and dialects affected American English less.

In any case, although both American English and English English have changed since colonial times, it is generally felt that in the case of accents at least, American English accents have probably changed less than English English accents in the last 400 years