Pro-business Republicans clashed with social conservatives. The State House clashed with the State Senate. The Texas Association of Business clashed with the Tea Party. White Republican lawmakers clashed with Hispanic Democratic legislators in the Capitol’s House chamber, cursing, shoving and threatening one another. And 28 months into his term, Mr. Abbott is facing a fundamental question: How conservative is conservative enough for the governor of a state that defines the right in America as much as California defines the left?

At least part of Mr. Abbott’s problem appears to be that he has yet to come up with an answer, allowing the cacophony of ideologies on the right and far right to answer for him.

“Why is he so hands-off?” asked Julie McCarty, the president of the Northeast Tarrant Tea Party in the Fort Worth area. “Is that what his dream was, to become governor of the greatest state in the nation so that he could sit out on everything?”

Mr. Abbott has faced challenges more daunting than reconciling feuding Republicans. Almost 33 years ago, in July 1984, when he was a 26-year-old law school graduate studying for the bar in Houston, he took a break to go for a jog with a friend. A large oak tree collapsed on his back, leaving him paralyzed below the waist and in need of a wheelchair for the rest of his life. He said that the accident had put the demands of his job in its proper context.

“It was actually just yesterday someone asked me, ‘How is it that you can stay so calm with everything going on in the Capitol?’” Mr. Abbott said during a recent interview. “And the answer is simple. And that is, when you have your life broken in half and realize that you’re going to be able to piece your life back together and overcome that, everything else in life is pretty easy to deal with.”