In his first comments, Biden, 81, said he was honored that voters chose to support him "at a time when Trump poses a greater threat than ever."
Trump said there was no time to celebrate and called Biden the worst president in American history.
"We will close the borders, we will do things no one has ever seen before, and we will make our country's economy the best in the world," Trump said.
Trump needed at least 1,215 Republican delegates to win the nomination, and Biden needed 1,968 delegates from his party.
Although primaries have not yet been held in many states, both Biden and Trump received enough votes to be officially nominated.
If Biden and Trump face each other, it will be the first time in nearly 70 years that two former presidents will meet in a rematch.
The 60th presidential elections of the United States will be held on Tuesday, November 5, 2024.
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