Whether military matters come to dominate the primary, in the remaining month before the Iowa caucuses, is likely to depend on events in Iraq and Iran — and perhaps in neighboring countries — and how severe and visible any ensuing clash with Iran turns out to be. Foreign affairs have so far played a strictly limited role in the Democratic race.

There have been major debate-stage duels over health care, taxation, immigration, criminal justice and gun control, but only glancing disagreements about the role of the United States abroad and the proper way to resolve American military engagements in the Middle East and Central Asia.

On Friday, much of the Democratic field proceeded with — and recommended — caution. Senator Amy Klobuchar of Minnesota called on the Trump administration to consult with Congress about a “strategy for preventing a wider conflict. ”And in North Conway, N.H., Pete Buttigieg, the former mayor of South Bend, Ind., called the Baghdad attack “an extremely provocative act,” noting that Presidents Barack Obama and George W. Bush had both considered but opted against attempting to kill General Suleimani.

“If we have learned nothing else from the Middle East in the last 20 years, it’s that taking out a bad guy is not a good idea unless you are ready for what’s coming next,” said Mr. Buttigieg, who referred several times to his own service in the military.

The possibility of a new and protracted conflict abroad could well reshape the general election, even beyond the Democratic race. Mr. Trump ran for president on a pledge to pull back the United States from foreign wars, drawing support from unconventional quarters for a Republican because of the perception that he would pursue an “America First” policy of relative isolationism and national self-interest.

But Mr. Trump had already drawn criticism from his Democratic rivals, and even within his own party, for presiding over a chaotic pullback from Syria, and the eruption of large-scale violence in Iran and Iraq could profoundly complicate his aim to seek a second term on a message of peace and prosperity. After Thursday’s attack, the Trump administration announced that thousands more troops would deploy to the region in anticipation of Iranian action.