This is plainly not the case. The grain of truth here is that certain linguistic features common to many dialects of 16th century British English have been preserved in most dialects of American English, whereas most dialects of British English have lost these features. (E.g., most American English dialects are rhotic.) But there is ample evidence that many aspects of American English have changed in the past few hundred years. Americans are clearly not speaking some kind of fossilized dialect from the 15th century.

By the way, the term "Old English" strictly speaking refers to this: https://en.wikipedia.org/wiki/Old_English. No-one is claiming that Americans speak this language.