“As you know, he believes Muslims should be banned from entering the country because of their faith,” she said. “What would that mean for a nation founded on religious freedom?” The struggle to replace Mr. Scalia has provided Mrs. Clinton with an opportunity to thrust the Supreme Court to the top of voters’ concerns.

In her speech on Monday, she noted that two Supreme Court justices will be older than 80 when the new president takes office, and she urged voters to “please make sure the court factors into your decision.”

“Whoever America elects this fall will help determine the future of the court for decades,” Mrs. Clinton said, listing a range of issues — from voting rights and abortion, to immigration, climate change and money in politics — that could reach the court in coming years.

Republicans, she said “are fighting hard to make sure the Supreme Court includes as many right-wing justices as possible.”

Fifty-three percent of Americans say the Senate should hold a vote on Mr. Obama’s nominee, while 42 percent said the Senate should wait until next year for the new president to nominate someone, according to the latest CBS News/New York Times poll. And while most voters have not yet formed an opinion about Mr. Garland, a vast majority — 68 percent — said that who sits on the Supreme Court is at least very important to them.

A spokeswoman for Mr. Trump did not immediately return an email request for comment on Monday. But some conservatives quickly rebuffed Mrs. Clinton’s speech, accusing her of politicizing the court. “If Hillary Clinton is criticizing you, you must be doing something right,” said Adam Brandon, the chief executive of FreedomWorks, an advocacy group aligned with the Tea Party. “The Senate is performing its constitutional responsibility” in delaying the process until a new president takes office.