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"I would never try and tell them get off the streets or anything like that, but we need to be together," Kasich told ABC News' chief White House correspondent Jon Karl in an interview. "The people of this country have spoken."
"There was concern about what would happen if the race was close and would people say it wasn't fair," he continued. "Look it was fair. There's nobody saying it wasn't fair. So now all of us take a deep breath. Now, all Americans take a deep breath."
Kasich made waves following the GOP presidential primary for breaking the Republican National Committee pledge by not supporting Trump. But now that the billionaire businessman is president-elect, Kasich says Americans of all political stripes should "give him a chance."
"Give him a chance and there'll be places where I agree with him and where I disagree with him, but that doesn't lead to any disrespect," Kasich said. "That's one of the things that's really important."
Kasich, who opted to vote for Arizona Sen. John McCain for president on Election Day, said he would not serve in the Trump administration if presented with the opportunity, but would provide his advice if asked.
"No, I'm going to be the governor for two years," he said. "I don't want to leave that position. If they call and want my advice or they want me to help them in some way. Just like when President Obama called on trade, you know. I disagree with the president most of the time, but if there's issues where I can help, where I can do something -- maybe even bridge some bipartisan feelings -- I'm more than willing to do that, but I don't expect to be in the administration."
Kasich, in attendance at today’s White House ceremony honoring the 2015-16 NBA champion Cleveland Cavaliers, said he spent a few minutes with President Obama and had a "good conversation" but they did not discuss the president’s meeting with Trump.
"We have a new president, and it' an exciting time to have a new president," he said. "We've got to put any of our differences aside and we've got to give him support."
"We can't have a country that's constantly fighting with one another," Kasich added. "Let's all say our prayers, let's wish him support and wish him the best for all of our people, for our kids."