Native Americans are also immigrants - but of approximately 13,000 years ago

Oh, Pooh! I knew that, Sweet Shanks!

FDR was jabbing at a group who considered themselves "THE" Americans ... and he used his opening to bring home the point that there are many "Americans."
Remember, sweetheart, those were the days when the Metropolitan Opera diva, contralto Marian Anderson, was denied the use of the DAR Hall for a concert. The DAR said the reason was not because Miss Anderson was a Negro (the term of the time) but because the Hall was too small to accommodate the expected audience.

Anyway, through the intercession of Mrs. Eleanor Roosevelt, the concert was held at the Lincoln Memorial with Miss Anderson on the steps just below the statue of The Great Emancipator, and many many thousands of people of all colors filling the mall below.
There are films of the event extant.
wow