"Based on the fact that the Democratic nominating process is totally rigged and Crooked Hillary Clinton and (Debbie) Wasserman-Schultz will not allow Bernie Sanders to win, and now that I am the presumptive Republican nominee, it seems inappropriate that I would debate the second-place finisher," Trump said in a statement.
Trump also said news networks "are not proving to be too generous to charitable causes," which he previously had said would be a goal of debating Sanders.
The two populist candidates had floated the idea of debating each other throughout the week. Sanders has been trying to debate Clinton ahead of the California primary on June 7, but she, too, declined.
Speaking to reporters in Los Angeles, Sanders said he was disappointed that Trump changed his mind.
"I hope that he changes his mind again. Mr. Trump is known to change his mind many times in a day," Sanders said. "Trump is a bully, he's a big tough guy. Well, I say to Mr. Trump, what are you afraid of?"
Trump's announcement on Friday is a far cry from a statement he made the day before, when he told supporters in Bismark, North Dakota, that he'd "love to debate Bernie."
"He's a dream," the real estate magnate said. "If we can raise for maybe women's health issues or something. If we can raise $10 or $15 million for charity, which would be a very appropriate amount."
"I understand the television business very well. I think it would get high ratings," Trump added.
The idea of the two debating surfaced on Wednesday night, when ABC's Jimmy Kimmel told Trump he had a question from Sanders' campaign, asking if Trump would be willing to debate the Democratic hopeful.
Trump joked, "How much is he going to pay me?"
"That could happen, I've been saying that should happen anyway," he added. "If he paid a nice sum for charity, I would love to do that."
Minutes later, Sanders responded on Twitter.
"Game on," Sanders wrote. "I look forward to debating Donald Trump in California before the June 7 primary."
Later in the day, Sanders' campaign manager Jeff Weaver told CNN's Wolf Blitzer that he hoped Trump doesn't "chicken out" after the two talked about the possibility of a one-on-one debate.
The two candidates reiterated their support for a debate throughout Thursday.