LUCCA, Italy — Days after President Trump bombed Syria in response to a chemical attack that killed children, Secretary of State Rex W. Tillerson said on Monday that the United States would punish those “who commit crimes against the innocents anywhere in the world.”

The declaration, given at a memorial to a Nazi massacre that killed many children, was among the first efforts by a top official to describe what seems to be a new Trump administration doctrine that encompasses instinctual and emotional responses to catastrophic world events.

Mr. Tillerson’s statement also seemed bound to intensify a growing rift between the Trump administration and Russia, where Mr. Tillerson is headed on Tuesday to confront Russia’s foreign minister, Sergey V. Lavrov, over the countries’ differences in Syria.

There had been some hope that Mr. Tillerson would meet President Vladimir V. Putin. But Russia announced on Monday that Mr. Putin would be unavailable — more signs of the Kremlin’s growing displeasure. Hopes in Russia for an enduring thaw in relations between the two countries, fed by Mr. Trump’s oddly positive comments about Mr. Putin during the campaign, have largely ended.