"You cannot have major changes in major programs affecting things like health care without including Democrats from the very beginning," Kasich, a Republican, said in an interview with Dana Bash on CNN's "State of the Union."
Republicans have pledged to repeal and replace Obamacare since it became law in 2010. But after several weeks of fraught debate and a lack of support from many within the GOP as well as the full Democratic caucus, Republican leaders pulled their bill on Friday.
Kasich said he thinks Trump would learn something from the defeat and possibly move to work with Democrats on compromise legislation.
"Look, he's going to learn from this, but you can't expect the executive to know everything," Kasich said, adding that he believes Trump's "instincts would have been to cut a deal and to bring the Democrats in."
After the bill was pulled on Friday, Trump gave brief remarks from the White House and blamed Democrats for the upset and said Obamacare would "explode." On Sunday morning, Trump took to Twitter say "Democrats are smiling" because conservative groups sunk his preferred legislation.
Trump's budget director, Mick Mulvaney, said Sunday he had "no idea" why he couldn't get the conservative House Freedom Caucus to support the bill.
"Is the Republican Party capable of governing?" Mulvaney said in an interview
on NBC's "Meet the Press."
During negotiations on the bill, conservative lawmakers balked at tax credits in the legislation, which they viewed as a form of entitlement, and ultimately sought
to repeal Obamacare's entire insurance regulation infrastructure.
In the face of the bill's failure, the Trump administration is moving on from health care for the time being, even though the President promised to repeal and replace the law, Mulvaney said. He added that the administration would get back to repealing Obamacare "when it breaks," which he insisted was inevitable.
"We've moved on to other things," Mulvaney said.
on the GOP bill from the Congressional Budget Office as well as an assessment
from S&P, a financial service and analysis company, have said Obamacare was not imploding.
Left, right and center
Seizing the moment in the wake of the GOP failure, Vermont Sen. Bernie Sanders has announced his intention to submit a bill establishing a single-payer health care system, in which the government pays the costs of coverage, which he has long advocated.
Republicans would be extremely unlikely to consider the legislation.
The progressive independent, who caucuses with the Democrats, said in a separate interview on CNN's "State of the Union" on Sunday that the country needed to "guarantee health care to all people as a right."
He also called the price of prescription drugs in the US an "absurdity" and called on the President to work with him to lower the cost of medicine.
"President Trump, come on board. Let's work together," Sanders said.
For his part, Kasich called for a centrist approach, saying those with extreme views on both sides of the aisle needed to be "marginalized" and that many Democrats would work with Republicans given the chance. Without bipartisan buy-in, "What you pass will never be sustainable," he said.
Kasich expanded Medicaid within his state under Obamacare, one of several Republican governors
to do so. He opposed the GOP bill and cited its approach to Medicaid as one reason he was not behind it.
Kasich, who campaigned against Trump in the Republican primary and refused to support him in the general election, said he didn't agree with the assessment that Trump wasn't invested enough in the policy details of the bill to be an effective advocate for it.
"The President's not supposed to be crossing all the T's and dotting all the I's," Kasich said.
Since his losing presidential campaign, Kasich has kept up his public profile somewhat, doing national interviews, maintaining a political organization and working on a book. When asked if these activities were a leadup to a potential run again in 2020, Kasich said, "No."
"I'm not really interested in running for political office again," Kasich said. "You don't close the door on anything, but I don't have my eyes on that."