Now, as the governor’s image has softened, his poll numbers have improved heading into a re-election race next year against the likely Democratic nominee, Ed FitzGerald, the executive of Cuyahoga County.

He still angers many on the left; he signed a budget in June that cut revenues to local governments and mandates that women seeking an abortion listen to the fetal heartbeat. Democrats see his centrist swing as mere calculation, a prelude to a tough re-election fight.

“This is someone who realized he had to get to the center and chose Medicaid as the issue,” said Danny Kanner, communications director of the Democratic Governors Association. “That doesn’t erase the first three years of his governorship when he pursued polices that rewarded the wealthy at the expense of the middle class.”

Ohioans earning in the top 1 percent will see a $6,000 tax cut under the latest budget passed by the Republican-led legislature, while those in the bottom fifth will see a $12 increase, according to Policy Matters Ohio, an independent research group.

The governor dismissed the notion that his Medicaid decision was political. “I have an opportunity to do good, to lift people, and that’s what I’m going to do,” he said. “You know what?” he added, using a phrase he utters before aiming a jab. “Let the chips fall where they may.”

The son of a mailman who grew up outside Pittsburgh, Mr. Kasich (pronounced KAY-sik) has said he didn’t meet a Republican until he arrived as a freshman at Ohio State. He has often showed an independent streak. He supported President Bill Clinton’s assault weapons ban while in Congress in 1994, and he teamed with Ralph Nader to close corporate tax loopholes.

In the interview in his office, he criticized a widespread conservative antipathy toward government social programs, which regards the safety net as enabling a “culture of dependency.”