He even tried out a new epithet for Mrs. Clinton: “heartless Hillary.”

Calling Mrs. Clinton and her husband, former President Bill Clinton, “heartless hypocrites,” he dared them to “let their bodyguards immediately disarm,” an apparent reference to their Secret Service protection.

“Let’s see how good they do,” Mr. Trump said. “Let’s see how they feel walking around without their guns and their bodyguards. In the meantime, nobody else can have the guns, right?”

Mr. Trump’s efforts to shore up his support among the N.R.A.’s more than five million members could help him in the Rust Belt states that he would need to carry to win the White House. But Mrs. Clinton must strike a more delicate balance on the issue: In the Democratic primaries against Senator Bernie Sanders of Vermont, who has a mixed record on gun control, she has taken an aggressive tack against firearm manufacturers and sellers.

In a general election contest with Mr. Trump, however, Mrs. Clinton would vie with him for the loyalties of white voters in a number of battleground states where support for gun rights runs deep. Indeed, in recent weeks, as she campaigned before largely white, working-class audiences in Appalachian and Great Lakes states, she has de-emphasized gun control and focused more on job creation and economic aid for financially struggling communities.

But on Saturday, Mrs. Clinton will speak at a dinner of the Trayvon Martin Foundation’s “Circle of Mothers” in Fort Lauderdale, Fla., a group offering support to women who have lost a child to gun violence. And she is expected to press the issue to win over voters in Los Angeles, Oakland and other California cities before that state’s primary on June 7.

Maya Harris, a senior policy adviser to Mrs. Clinton, dismissed Mr. Trump’s attacks on Friday, saying he was “peddling falsehoods.”