"I'm rooting for him to get it together," Kasich said on CNN's "State of the Union." "We all are."
The Republican governor said he was trying to be a "responsible voice" and offered hope that Trump -- whom Kasich unsuccessfully challenged in the 2016 Republican presidential primary and whom he has repeatedly criticized -- would grow from experience in office.
"I hope we're going to have stability," Kasich said. "The President is going to learn from these episodes, and we're going to do better. That's what I hope is going to happen."
Asked about the recent ouster of Steve Bannon as White House chief strategist, Kasich said the staff shakeups were up to the President and not a cure-all to Trump's difficulty notching accomplishments on his agenda.
"The changes have to stop, and we have to have a team," Kasich said. "You can't keep putting new people in the lineup and think you're going to win a world championship."
Kasich has criticized Trump on a wide range of issues, recently calling
Trump's response to racially charged violence in Charlottesville, Virginia, "pathetic." But in his comments Sunday, Kasich said people should not "look backwards" and tempered his criticism of the President.
Kasich said ahead of Trump's hotly anticipated speech in Phoenix on Tuesday that he hopes those around Trump will get the President to bring people together instead of stoke division.
"When you go, try to use that as an opportunity to say something that's going to bring people together," Kasich said.
Trump's speech Tuesday follows his comments last week attacking Republican Sen. Jeff Flake and suggesting
he was mulling a pardon for former Sheriff Joe Arpaio, who was found guilty of criminal contempt of court stemming from a racial-profiling case.
Kasich weighed in at length on the foreign policy front, offering specifics about what he would do in two key areas facing Trump.
Kasich criticized the idea of nation-building in Afghanistan, where the US led an invasion following the September 11 attacks, and said, as he did on the campaign trail, it was time for the US military to withdraw.
"I think we need to begin to leave there," Kasich said.
He advocated, however, maintaining the option to conduct strikes there in response to intelligence.
Trump's team has been working on a new strategy for US in Afghanistan and Pakistan, and Secretary of Defense James Mattis has said Trump was presented with a range of options, from full withdrawal to an increase in troop levels there. On Saturday, Trump tweeted he had made a decision, and Mattis on Sunday confirmed the development but said Trump would be the one to make the decision public.
Kasich also staked out a position perhaps even more hawkish than Trump's on North Korea. He said that if he were President, he would reserve the right to use a pre-emptive strike on Pyongyang to destroy its ability to hit the US with a nuclear weapon and make that option clear to China.
The Ohio governor argued the threat to attack North Korea would either spur China to negotiate the end of the North Korean nuclear threat to the US -- or the US would attack the isolated nation of millions.
"It needs to be made very clear to the Chinese: This is in your hands, and if you don't want to do anything, you're going to have to live with the result," Kasich said.
The Trump administration has repeatedly called for Chinese cooperation and the de-nuclearization of North Korea, and Trump warned earlier this August he would respond to North Korean threats with unprecedented "fire and fury."
Ahead of joint US-South Korean military exercises this week, Pyongyang said
Sunday that the US would not be able to avoid a "merciless strike" from North Korea.