Mr. Trump’s letter, which was obtained by The New York Times and later posted by the union, was dated Feb. 4 and addressed to Ms. Carteris. In it, he blasted the organization, asserting that it had “done little for its members, and nothing for me.”

“I no longer wish to be associated with your union,” Mr. Trump wrote. “As such, this letter is to inform you of my immediate resigning from SAG-AFTRA.”

Mr. Trump also used the letter to enumerate his own acting accomplishments and argue that he had helped the cable news business thrive. He wrote that he was “very proud of my work on movies such as ‘Home Alone 2,’ ‘Zoolander’ and ‘Wall Street: Money Never Sleeps’; and television shows including ‘The Fresh Prince of Bel-Air,’ ‘Saturday Night Live,’ and of course, one of the most successful shows in television history, ‘The Apprentice’ — to name just a few.’’

Responding to Mr. Trump’s resignation, SAG-AFTRA offered a simple two-word statement jointly attributed to Ms. Carteris and David White, the union’s national executive director on Thursday afternoon: “Thank you.”