Trump wrote on Twitter: "The Theater must always be a safe and special place. The cast of Hamilton was very rude last night to a very good man, Mike Pence. Apologize!"
"Our wonderful future V.P. Mike Pence was harassed last night at the theater by the cast of Hamilton, cameras blazing. This should not happen!" Trump added.
Pence was greeted with a mix of boos and cheers as he entered the Richard Rodgers Theater in New York on Friday night to watch a performance of the highly acclaimed musical about the country's founding fathers, played by a racially diverse cast.
After the show, Brandon Victor Dixon, who plays America's third vice president Aaron Burr, read a statement directed at Pence while standing in front of the cast in full costume.
"We, sir - we - 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," Dixon said. Pence had already begun to leave his seat as Dixon began his remarks, videos posted on social media showed.
"We truly hope that this show has inspired you to uphold our American values and to work on behalf of all of us," Dixon said as audience members cheered and clapped.
A show spokesman said Pence stood in the hallway outside the entrance to the auditorium and heard the full remarks, the New York Times reported.
The incident in the theater happened the same day that Trump picked three conservative loyalists to lead his national security and law enforcement teams, setting his administration up to take a hard line confronting Islamist militancy and curbing illegal immigration.
"Hamilton" is a musical biography of Alexander Hamilton, who rose to become the right-hand man of General George Washington, as well as a key figure in the creation of the U.S. financial system and the creator of the U.S. Coast Guard. He was killed in an 1804 duel with Burr.
The actor Dixon fired back at Trump's allegations of harassment.
"@realDonaldTrump conversation is not harassment sir. And I appreciate @mike_pence for stopping to listen," he tweeted in response to Trump's Tweet. The show's award-winning creator, Lin-Manuel Miranda, said he was proud of the cast.
"Proud of @HamiltonMusical. Proud of @BrandonVDixon, for leading with love. And proud to remind you that ALL are welcome at the theater," Miranda wrote on Twitter.