Addressing a packed crowd of over 50,000 at the Wells Fargo Center in Philadelphia, Mrs Clinton said that "America's strength doesn't come from lashing out" when faced with hostility.
"Imagine him in the Oval Office facing a real crisis," the Democratic candidate said of Mr Trump.
"A man you can bait with a tweet is not a man we can trust with nuclear weapons.
"He loses his cool at the slightest provocation – when he’s gotten a tough question from a reporter, when he’s challenged in a debate, when he sees a protestor at a rally.
"Donald Trump can’t even handle the rough-and-tumble of a presidential campaign.
"If we're serious about keeping our country safe, we can't afford to have a President who's in the pocket of the gun lobby."
Mr Trump has repeatedly stated his intention to uphold the Second Amendment, the right to keep and bear arms, and has said gun ownership makes the US a safer place.
Mrs Clinton also made her feelings known about Mr Trump's campaign pledges and questioned his patriotism.
"He talks a big game about putting America first. Please explain to me what part of America first leads him to make Trump ties in China, not Colorado," she said.
"Donald Trump says he wants to make America great again. Well, he could start by actually making things in America again."
Criticising Mr Trump's business pedigree, Mrs Clinton added: "That sales pitch he’s making to be your president? Put your faith in him – and you’ll win big? That’s the same sales pitch he made to all those small businesses. Then Trump walked away, and left working people holding the bag."
Mrs Clinton's comments were met with loud applause and cheers from the Democratic contingent.
The speech by the first woman to be nominated as a presidential candidate by a major party in America ended this year's Democratic National Convention.
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