In an interview with CNN's Jake Tapper on "State of the Union," Bush hit the two for opposing President Barack Obama's request for authorization to use military force after Syrian President Bashar al-Assad was accused of using chemical weapons against his own people.
Bush said if the two thought Obama's request was too narrow, they could have pushed for tougher language.
"Now everybody's talking tough, and that's not the sign of leadership ... You need to think these things through, and if it's not popular, persuade people toward your cause. That's what leadership is all about," Bush said.
He said Rubio and Cruz have only staked out stronger positions in the wake of ISIS attacks in Paris and San Bernardino, California.
"They could have led the charge to create a strategy to deal with ISIS and Assad, and they didn't, because it was not popular at the time. And then conditions change," he said.
Bush added: "I am saying that because when they had a chance to support a more engaged America in Syria that could have saved lives ... they decided no because at the time, the Republican voters were opposed to it."
Bush had tough words for Republican front-runner Donald Trump, saying he's insulted too many people -- including women, Hispanics, John McCain and the disabled.
"He's not the strong horse. He's insecure and he's weak," Bush said of Trump.
Over the weekend, after Barbara Bush recorded an ad touting her son's decision to run for president as more than "talking about how popular they are or how great they are" -- a clear jab at Trump -- Trump responded that Bush "desperately needed mommy to help him."
Bush's campaign countered with an image of Barbara Bush wearing football shoulder pads.
But on Sunday, Bush turned the talk of his mother into an attack on Trump, criticizing Trump's comment on Twitter that "mom can't help you with ISIS."
"Donald Trump has no clue how the world works. There's no negotiating with ISIS," Bush said.
He also said he hopes to see his brother, former President George W. Bush, on the campaign trail -- though that won't happen before Iowa's caucuses on February 1.
"No, I don't think it is because of schedules, but I hope he is going to be involved, and the campaign is working with him on that," Bush said.
A Bloomberg independent run
Bush said he doesn't believe Trump will emerge as the GOP nominee, saying that "the party process is going to work far different than what the pundits are saying. I really believe that."
But if Trump does win, and former New York City Mayor Michael Bloomberg mounts an independent campaign, Bush said he has big differences with Bloomberg.
"We have differing views, and I want to be the conservative candidate to lead the conservative party into the general election," Bush said.
Flint's water crisis
Bush defended Michigan Gov. Rick Snyder, a Republican who has faced intense criticism over his handling of the water crisis in Flint.
"Instead of saying, 'the dog ate my homework, it's someone else's fault,' once it became clear, he's taking the lead now," Bush said of Snyder.
"It does point out that we have a 20th century regulatory system on a 21st century world. Someone needs to change how we go about Washington's role in this so there's more accountability and more transparency," he said.