A Speaker Boehner would also face the challenge of keeping a diverse mix of House Republicans in check, including a new class of political rookies and Tea Partiers, some of whom have been sharply critical of party leadership.
But the Ohio Republican, who said patience is the greatest lesson his parents taught him, said he doesn't see the cast of Tea Party candidates as a threat to his ability to build a coalition.
"You learn to deal with every character that walks in the door," Boehner said of his experience growing up in his father's bar. "Trust me, all the skills I learned growing up are the skills I need to do my job."
Tea Partiers "are the most down to earth, ordinary people who want the same thing for our country as Republicans, Democrats," he said. "They see all the spending and all the debt and the growing size of government as a threat… as do I."
Does Boehner believe the House Republican leadership team should include a Tea Party-backed member? "Whoever the members elect, I'm going to serve with and serve with successfully," he said.
Sawyer also pressed Boehner on whether he would be uncompromising on a full, permanent extension of the Bush tax cuts, which the Congress will consider when it returns later this month.
Obama and Democrats have called for a permanent extension of the cuts for the middle class, those Americans making less than $250,000. Many Republicans want the cuts preserved for all Americans.
"I believe we have to extend all the current tax rates for all Americans," Boehner said. "It begins to reduce the uncertainty." Boehner would not clarify whether extending the cuts means making them permanent.
Boehner, 60, one of 12 children from an Ohio Roman Catholic family, grew up working in his father's bar and later became president of a plastics company. He was first elected to Congress in 1990.
"We're savers, not spenders," he said of the Boehners. "I wouldn't describe myself as a penny pincher. But debt is just not something that makes me comfortable, and this national debt makes me really uncomfortable."
If elected by his party, Boehner would become the 61st speaker of the U.S. House and third in the line of presidential succession.
"I'm calm, confident and know that if I listen to the American people and be myself, this job will be just fine," he said.