There is only one song written by a US Vice President. "Melody in A Major" was written in 1911 by Charles Dawes, who became vice president under Calvin Coolidge. Of course that is not the famous part. Words were added in 1951 and it became a hit when it was re-recorded in 1958. If you were not listening to music back then, you can stop here, but if you were - go to the link to find the answer.
LINK Hint, the singer's first name was Tommy.