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Gov. John Kasich of Ohio on Tuesday cautioned against monitoring Muslim-Americans after the attacks in Brussels, saying that such a step would create division and harm the country’s ability to gather intelligence.

His comments came after one of his rivals, Senator Ted Cruz of Texas, called for law enforcement “to patrol and secure Muslim neighborhoods before they become radicalized.”

Mr. Kasich told reporters in Minneapolis, “We are not at war with Islam; we’re at war with radical Islam.”

“In our country,” he said, “we don’t want to create divisions where we say, ‘O.K., well your religion, you’re a Muslim, so therefore we’re going to keep an eye on you.’” He added that “the last thing we need is more polarization.”

“Frankly, for those who want to preserve Islam as a religion that is not at war with the West, we alienate them, how are we supposed to ever get the information we need?” he asked.

Mr. Kasich hopes that his foreign policy experience, gained from his 18 years in Congress, will help him stand out compared with Mr. Cruz and Donald J. Trump.

Setting up a contrast with Mr. Trump, Mr. Kasich also said that he would “make every effort I could to strengthen the NATO alliance.” On Monday, speaking to the editorial board of The Washington Post, Mr. Trump questioned the United States’s involvement in NATO.

Mr. Kasich also said President Obama should cut short his visit to Cuba in order to deal with the unfolding situation in Europe.

“If I were in Cuba right now, the last thing I would be doing is going to a baseball game,” he said, referring to Mr. Obama’s plans to watch the Tampa Bay Rays play the Cuban national team. “I would be going back to Washington.”