"That Trump echoes the attacks of Michael Moore and the fringe Left against my brother is yet another example of his dangerous views on national-security issues," Bush wrote.
It's a continuation of the feud that started last week when Trump blamed Bush's brother, former President George W. Bush, for the September 11, 2001, terrorist attacks.
The former Florida governor has said his brother kept the country safe after the unforeseen attacks -- but Trump has charged that it was also up to George W. Bush to prevent them.
"Let's be clear: Donald Trump simply doesn't know what he's talking about. And his bluster overcompensates for a shocking lack of knowledge on the complex national-security challenges that will confront the next president of the United States," Bush wrote.
Bush said Sunday in an interview with CNN's Jake Tapper on "State of the Union" that he has "grave doubts"
about Trump as commander-in-chief.
"He's not taking the responsibility, the possibility of being president of the United States really seriously. For him, it looks as though he's an actor playing a role of the candidate for president. Not boning up on the issues, not having a broad sense of the responsibilities of what it is to be a president," Bush said. "Across the spectrum of foreign policy, Mr. Trump talks about things as though he's still on 'The Apprentice.'"
Trump, meanwhile, claimed that he could have prevented the attacks.
In an appearance on "Fox News Sunday," Trump said that because he's "extremely, extremely tough on immigration," the attackers wouldn't have been in position to commandeer U.S. flights.
"So there's a good chance that those people would not have been in our country," Trump said.
"I'm not blaming George Bush," Trump said. "But I don't want Jeb Bush to say, 'My brother kept us safe,' because September 11 was one of the worst days in the history of this country."