As Mr. Pence stood by the exit doors, Mr. Dixon said, “We, sir, are the diverse America who are alarmed and anxious that your new administration will not protect us — our planet, our children, our parents — or defend us and uphold our inalienable rights, sir. But we truly hope this show has inspired you to uphold our American values and to work on behalf of all of us. All of us.”

The “Hamilton” episode is the first major collision between the two Americas brought into sharp relief by the Nov. 8 election.

The values and politics championed by the cast are in conflict with the remarks and actions of Mr. Trump, who has called for deporting undocumented immigrants, has declined to forcefully denounce expressions of bigotry among his allies and has so far appointed only white men to major cabinet positions. He has also pledged to change libel laws and sue news media organizations whose coverage he does not like, and he has demonized street protesters who have criticized him.

Although Mr. Pence listened to the entreaty inside the Richard Rodgers Theater on Friday and then stepped onto the sidewalk smiling, Mr. Trump took time out of preparing his new cabinet on Saturday to rally his supporters. Around 9 a.m. he wrote two Twitter posts saying the “Hamilton” actors had harassed Mr. Pence and had been “very rude last night to a very good man.”

“Apologize!” Mr. Trump wrote.

Mr. Dixon, who read the statement after playing the nation’s third vice president, quickly replied with a post of his own: “@realDonaldTrump conversation is not harassment sir. And I appreciate @mike_pence for stopping to listen.”

Mr. Trump lashed out at the show, the most acclaimed Broadway production in years, at a time of demonstrations against his coming presidency. Those include frequent street protests outside Trump Tower along Fifth Avenue in Manhattan, which is less than a mile from the theater showing “Hamilton.” The president-elect has both castigated the protesters and, after being chided for doing so, praised them for their passion. Advisers, however, say he has been frustrated by the suspicion and hostility that the demonstrators and other Americans continue to hold about his election.

Mr. Trump’s escalation of the “Hamilton” matter also came a day after his $25 million settlement of a lawsuit against Trump University, leading some critics to claim online that he was trying to create a distraction amid negative attention to his surrender on a legal case he had vowed to fight.