Mr Biden’s letter said Democratic voters in the primaries have “spoken clearly and decisively” that he should be the party’s nominee.
“Do we now just say this process didn’t matter?” the letter said. “That the voters don’t have a say… I decline to do that. How can we stand for democracy in our nation if we ignore it in our own party? I cannot do that. I will not do that.”
Mr Biden also phoned Democratic donors on Monday. One source familiar with the call told CBS News, the BBC’s US partner, that the president said his strategy for the second debate against Trump in September will be “attack, attack, attack”.
Several congressional Democrats have called for Mr Biden to drop out, but late on Monday, several others rallied round the embattled president.
Left-wing New York lawmaker Alexandria Ocasio-Cortez told reporters: “The matter is closed. Joe Biden is our nominee.
“He is not leaving this race. He is in this race and I support him.”
Ayanna Pressley of Massachusetts, Maxine Waters of California, and Frank Pallone of New Jersey echoed that support for Mr Biden.
Earlier in the day, Adam Smith of Washington state became the sixth member of Congress to publicly urge Mr Biden to quit.
“We need a stronger messenger,” he told the BBC, as he panned Mr Biden’s debate performance.
“The president was completely incapable of doing something that any sort of relatively novice debater should have been able to do, and it hasn’t gotten better since then,” he added.
On Sunday, the Democratic minority leader in the House of Representatives, Hakeem Jeffries, held a group call in which several congressmen were explicit in urging Mr Biden to step aside, according to US news outlets.
They reportedly included Jerry Nadler of New York, Mark Takano of California, Joe Morelle of New York and Jim Himes of Connecticut.