Gov. John Kasich of Ohio said on Sunday that some “small steps” should be taken on gun control, addressing background checks and mental illness. Gov. Phil Scott of Vermont said in a statement that his administration would review its procedures and policies on gun safety.

Neither suggested that change was imminent. Their remarks recalled other murmurings of change that ultimately fizzled. Congress has been marked by intransigence on gun legislation, including a failure last year to ban so-called bump stocks, an accessory that the gunman in the October shooting in Las Vegas used to transform his semiautomatic rifles to mimic automatic weapon fire. That shooting left 58 people dead and wounded hundreds.

Students from Parkland and across the country have organized protests and marches to urge Mr. Trump and lawmakers to act. About 100 people gathered in front of the White House on Monday to rally for stricter gun control. Seventeen students, representing the number of people killed in Florida, lay on the ground in protest. Parents, teachers and friends joined them on the pavement, and high school students stood in clusters in 40-degree temperatures, chanting “Enough is enough!” and waving signs that read “Am I next?”

“I’ve been afraid to go to school since the shooting in Florida,” said Maya Galanti, a 12-year-old from Bethesda, Md., who attended the rally with her mother and two siblings. “Those students thought they were having a normal day, and we have the same chances of getting shot as they did.”

Students from the Parkland area have also lashed out at Mr. Trump on Twitter, and some were incensed when he suggested in a tweet that the shooting had occurred because the F.B.I.’s resources had been diverted to the Russia investigation.

“Seventeen innocent people were brutally murdered at my school, a place where they should have felt safe,” one student wrote. “Their lives were gone in an instant. You are the president of the United States, and you have the audacity to put this on Russia as an excuse.”

The legislation last fall was considered a modest step toward a bipartisan compromise on gun safety.