"The New World" is a phrase predating by hundreds of years the establishment of the United States. As such, accurate information on the history of a subject like theater is a little scarce, and designations of "the first" somewhat tenuous. That said, two histories of theater in North America provide logical answers to the question of the first record of a script in "the New World."
Early English-language settlements were familiar with and known to attend productions of leading playwrights of the time, especially productions of the works of William Shakespeare. Theater has been around since ancient times, so it is not surprising that the early settlers to North America would exhibit an interest in performances of drama and comedy. An early 20th century theater critic and journalist, Arthur Hornblow, wrote in his history of the American theater:
"In 1690 Harvard students gave a performance at Cambridge, Mass., of Benjamin Colman's tragedy Gustavus Vasa, the first play written by an American acted in America, of which we have any knowledge."
That bit of information is interesting, until one encounters the controversy surrounding the authorship of Gustavus Vasa, which was based on the reign of a 16th century Swedish king. Hornblow's attribution of the play to Benjamin Colman may or may not be accurate, as others have attributed the play -- and there may have been more than one -- to Alexis Piron and, more popularly, to American Henry Brooke. Irrespective of authorship, a credible answer to the question is Gustavus Vasa.
There is less uncertainty regarding the identification of the first American comedy written for the stage. The Contrast, written by prominent jurist-turned-playwright Royall Tyler, was first produced in 1787. Tyler's protagonist, Jonathan, was considered the quintessential American, or Yankee, in a story that, as one historian described it, defined and glorified "a distinctive American character embodying innocence, virtue, and sincerity." [http://www.jstor.org/stable/25070860?seq=1#page_scan_tab_contents]
As Tyler's play was truly American, it cannot be considered the first script written in the New World. That honor, controversy or uncertainty surrounding its authorship aside, should go to Gustavus Vasa.