COLUMBUS, Ohio — In his grand Statehouse office beneath a bust of Lincoln, Gov. John R. Kasich let loose on fellow Republicans in Washington.

“I’m concerned about the fact there seems to be a war on the poor,” he said, sitting at the head of a burnished table as members of his cabinet lingered after a meeting. “That if you’re poor, somehow you’re shiftless and lazy.”

“You know what?” he said. “The very people who complain ought to ask their grandparents if they worked at the W.P.A.”

Ever since Republicans in Congress shut down the federal government in an attempt to remove funding for President Obama’s health care law, Republican governors have been trying to distance themselves from Washington.