Is it true that Thomas Paine first coined the phrase, "The United States of America?"

Yes. Previously it was claimed that American Crisis II was the source of this name. There are at least three private letters who use the language "United States of America", but two of them use "united" (small "U") as an adjective. However, writing as "Republicus" on June 29, 1776 Paine made the first public declaration calling for the country to be named the "United States of America".