Kasich added: “The Western world and the Arab world need to get serious about stopping this kind of radicalism. If not, it will continue to spread and just bombing is imposing the status quo at best.”
Kasich cautioned, however, that he is recommending targeted and multinational strikes, rather than a drawn-out conflict against the Islamic State militants in Iraq.
"It is probably something that can be addressed without an extended affair and without nation-building or any of that,” he said.
Kasich's comments appear to position him between his party's two poles on foreign policy, blending a hawkish impulse with an appreciation for the war-weariness that has seeped into some quarters of the GOP since George W. Bush's presidency.
In the interview, Kasich acknowledged that he is not aligned with any particular wing of the Republican Party.
“It's pretty hard to peg me,” he said.
Speaking a day ahead of his visit to South Carolina, Kasich also touted his "18 years on the Armed Services Committee" while in Congress to underscore his knowledge of military and global matters.
"I was dealing largely with military operations, as well as how you use them and when you use them," Kasich said. "I have been able to travel and get a pretty good understanding of a lot of things."
In recent weeks, Kasich said he has had conversations that have touched on foreign policy with former House speaker Newt Gingrich, former New Hampshire senator John E. Sununu and former Pennsylvania congressman Bob Walker, among others.
Hardly a consistent backer of increased military spending during his time on Capitol Hill, Kasich said he understands the concerns some Republicans have about various Pentagon projects, and recalled how he openly debated U.S. involvement in foreign crises, going back to Ronald Reagan's decision to send Marines to Lebanon in the early 1980s.
"I voted against going to Lebanon as a young congressman and then for the Gulf War," he said. "I don't like the idea of getting involved in civil wars and I've always been a military reformer who is interested in special forces and using technology in better ways.”
Turning to domestic issues, Kasich insisted he remains undecided on whether to enter already crowded race for the Republican presidential nomination, but said he does have a growing desire to go on the road to discuss his record and the need for fiscal discipline within the federal government.
Whether these travels will eventually morph into a campaign is unclear, but Kasich is far from ready to rule it out and said he is actively thinking through his own political future.
"I want to have a national role so I can tell people about how things worked in Ohio and how others can learn from it," said Kasich -- especially his efforts to "bring people out of the shadows."
Kasich has angered some conservative Republicans with his policies, and his emphasis on compassion over ideological purity. He has increased state spending for social programs and accepted an expansion of Medicaid in Ohio under the Affordable Care Act.
Still, Kasich, a garrulous and populist figure who has long harbored presidential ambitions, firmly believes his identification with the “little guy” and his political success in the Rust Belt — a general-election battleground — gives him ample room and time to explore, and to potentially compete with former Florida governor Jeb Bush and others who have been pitching themselves as candidates who can broaden the GOP’s appeal.
"The whole country is polarized -- rich versus poor, black and white, Republican and Democrat," he said. "The country is in a position where it’s like we're all divided. I don’t know what it’s going to take to move the system, but it's about having a vision and how you explain what needs to be done."
Kasich’s two-day visit to Columbia, South Carolina's state capital, will feature a flurry of private and public appearances, beginning Wednesday evening with cocktail reception for state Republican leaders and closing with a news conference on Thursday following a huddle with state lawmakers.
Balanced Budget Forever, a non-profit advocacy group run by Kasich associates, is funding the trip, which is being billed as part of his tour around the country to promote a constitutional amendment to balance the federal budget. The cause has for years animated Kasich, who formerly chaired the House Budget Committee.
“Instead of young people talking about Justin Bieber or how bad the Grammys were, maybe we could get people talking about Washington, Madison, and Monroe, and have a renewal of American history,” Kasich said of a possible constitutional gathering to consider a balanced-budget amendment. “The more and more states that sign on, the more hopeful I get.”
But by appearing in South Carolina, Kasich is sending a not-so-subtle signal to donors and activists that he is openly mulling a second campaign for the White House, nearly 16 years after his first attempt fizzled, when he pulled out months before the 2000 presidential primaries began.
Should he jump into the 2016 race later this year, Kasich shrugged off what would be steep challenges on fundraising, where his rivals are now outpacing him.
"I am not antsy," he said. "I'm the governor of Ohio," unlike in 1999 when he was a "former congressman running from the 12th district of Ohio."