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

Yes. Previously it was claimed that American Crisis III was the source of this name. There are at least two private letters who use the language “united States of America”, but they 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”, which he knew appeared on the Declaration of Independence soon to be released.