Bush also said he has little confidence in Trump's ability to appropriately handle America's nuclear weapons.
"I have grave doubts, to be honest with you," Bush told CNN's Jake Tapper in an interview airing Sunday on "State of the Union."
"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.'"
Bush said Trump's proposed plan of hoping ISIS removes Syrian President Bashar Assad from power and then Russia taking on ISIS is like "some kind of board game and not a serious approach."
"This is just another example of the lack of seriousness. And this is a serious time. We're under grave threats again, and I think we need a president with a steady hand," Bush said.
Bush again dismissed Trump's suggestion that George W. Bush was responsible for 9/11 because it occurred during his presidency.
"My brother responded to a crisis, and he did it as you would hope a president would do. He united the country, he organized our country and he kept us safe. And there's no denying that. The great majority of Americans believe that," he said.
"And I don't know why he keeps bringing this up. It doesn't show that he's a serious person as it relates to being commander in chief and being the architect of a foreign policy," Bush added.
Bush said his defense of the 43rd president's response to the attacks isn't just because he's his brother.
"I mean, so next week, Mr. Trump is probably going to say that FDR was around when Japan attacked Pearl Harbor. It's what you do after that matters. And that's the sign of leadership," he said. "You don't have to have your last name named Bush to be able to understand that."
Despite Trump leading the polls, Bush predicted that support for him will eventually fade.
"I don't think Trump is going to win the nomination. I think we're going to have a nominee that will unite the party," Bush said.
He also offered praise for Democratic front-runner Hillary Clinton's debate performance, though he argued the policies she supports are bad for the country.
"She did a good job in the debate, for sure," he said. "She's a smart person, no doubt about that. But every chance she had to lay out a different approach than the one we're on now, she actually doubled or tripled down on it: more taxes, more regulation, more creating barriers on people's ability to rise up."