It came as Congress certified the Electoral College vote that gave him victory in the election.
During proceedings, Maxine Waters, the US Representative of California’s 43rd congressional district, shouted: “I do not wish to debate, I wish to ask: Is there not one US senator who will join me in this objection?”
Vice President Joe Biden had to reject repeated objections by Democrats to the official declaration, as well as deal with protesters at the congressional ceremony.
Mr Biden made several calls to the sergeant-at-arms to expel the protesters as he tried to read out the results of November’s vote.
Mr Trump got 304 electoral votes, compared with the 227 won by his Democratic challenger, Hillary Clinton, according to the vote tally.
The electoral votes were opened before a joint session of Congress in what is a formality for most elections.
Members of the House of Representatives objected to the electoral tally in states including Alabama, Florida, Michigan, Texas, Mississippi and the Carolinas.
Some members complained about long lines at polling stations while others highlighted concerns over Russian attempts to influence the result in Mr Trump’s favour.
“I object because people are horrified by the overwhelming evidence of Russian interference in our election,” said Representative Barbara Lee of California, before being overruled.
Mr Biden denied each of the objections and said: “It is over.”
None of the representatives had the backing of a senator, which would have allowed them to suspend the joint session and allow the House and Senate to meet separately to debate the objections.
Mr Trump, who will take office on 20 January, won the vital Electoral College vote that decides the winner, but Mrs Clinton won more votes overall.