Bush, a former Florida governor, also blasted Trump on Saturday during Ozy Fest, a combination of policy-related symposiums and musical acts hosted by the digital news magazine in New York's Central Park.
"If your opponent does things that you, your head explodes on, if Barack Obama did something as it's related to Russia, you say 'this is outrageous,' all this stuff, then when your guy does the same thing, have the same passion to be critical," Bush said.
When asked whether Republicans were afraid of Trump, he continued: "Does everything have to be a political calculus? 'Oh, my god, if I say something, there will be an opponent, and there will be a third-party interest group come and give money to my opponent. Oh, my god. Oh, my god. Oh, my god.' This is not what public services should be about."
Bush, who did not vote for Democrat Hillary Clinton or Trump in the 2016 election, has criticized Trump repeatedly since the president took office in January.
Bush addressed Trump's way of governing Saturday.
"You get disciplined when your team says, 'No, Mr. President, let's stay focused on these policy objectives,'" Bush said. "Don't disparage people, don't go after Mueller, don't say you're going to pardon yourself or whatever. Don't do all that. Govern."
Independent counsel Robert Mueller is investigating the possible collusion.
"He unleashed five tweets today about stuff that jeopardizes his legal situation, insults somebody, goes back to the 2016 election, none of which is relevant to getting tax relief done, regulatory reform done," Bush said about Trump. "This is going to be a long ride between 2018 in January, much less 2020."
Bush also urged more civility.
"Now, strength is measured by how you insult people," the ex-presidential candidate told the crowd. "Reject it, because we're never going to solve problems that way."
He also said: "The president is the prime minister and the king. He or she, eventually, will be the symbol of the country, and also the prime minister responsible for making government work," Bush said. "And right now, our president doesn't view that job as important. Look at history. History is important. When presidents inspire us, we do better. And that's what we need to get back to."
Bush noted that Trump was registered at one time as a Democrat in the district where Ozy Fest was being held.
Bush, who appeared with businessman Mark Cuban onstage, ruled out another presidential bid.
"My focus now, our focus now, is to get a Democratic Congress elected in 2018," Biden, 74, said to cheers. "There's a lot of talent in the Democratic Party. My hope and expectation is that some of the younger folks are going to come up and fill the void."
Without naming them, Jeb Bush calls out Republicans who were mad at Obama over Russia, but have been silent on Trump and Russia pic.twitter.com/4fuZG8hHDV- Max Tani (@maxwelltani) July 22, 2017