The House Speaker, Paul Ryan, urged delegates to unite behind Mr Trump, a day after splits in the party were evident as the convention opened.
The Trump campaign also faces accusations a speech by Mr Trump’s wife Melania on Monday was plagiarised.
Tuesday’s speakers focused almost exclusively on attacking Hillary Clinton, the likely Democratic nominee.
New Jersey Governor Chris Christie, a former prosecutor, held a mock trial for Mrs Clinton as the crowd chanted “lock her up”.
Mr Christie and others criticised Mrs Clinton’s use of a private email account while she was serving as secretary of state.
An FBI investigation said she was “extremely careless” but found her actions didn’t warrant criminal prosecution.
However, Mr Christie and and the crowd disagreed as Mr Christie repeatedly yelled “guilty”.
He said she has “selfish, awful judgement” and was to blame for various foreign policy problems in Libya, Syria and elsewhere.
Mr Trump is expected to accept the nomination on Thursday.
His children played a prominent role on Tuesday, standing with the New York delegation as he was declared winner and delivering remarks.
Mr Trump youngest daughter, Tiffany Trump, whose mother is former model and dancer Marla Maples, told some personal stories about her father.
She recalled scribbling notes in her school report cards and how excited she becomes when introducing him to her friends.
Her father is a “natural-born encourager” who has motivated her to work hard, she said.
His son Donald Trump Jr described him as his best friend and role model.
“When people tell him it can’t be done, that guarantees it will get done,” he said of his father.
He said Mrs Clinton was a risk the US could not afford to take and that “if she were elected, she would be the first president who can’t pass a background check”.
Mr Trump addressed the audience via a live-stream and said the nomination was an honour.
“This is a movement, but we have to go all the way,” he said. “This is going to be a leadership that puts American people first.”
Parts of Donald Trump Jr’s speech used segments of an article that had already appeared in the journal The American Conservative.
But the article’s author, FH Buckley, said it was not plagiarism, as Mr Buckley himself acted as one of the family’s speechwriters.
After speaking at the convention on Monday, Melania Trump, the mogul’s third wife, faced accusations that a portion of her speech plagiarised Michelle Obama.
But Mr Trump’s team said she used “common words” and accused the Clinton campaign of stoking the controversy.
On Monday, anti-Trump delegates failed in their final push to block his nomination.
A vote that would have allowed delegates to back a candidate of their choice was quashed when three states reportedly backed out.
Some said the Republican Party officials had sabotaged their efforts on purpose.