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."
