Gov. Rick Scott and top state lawmakers proposed on Friday the most significant move toward gun control in Florida in decades, backing new limits that defy the National Rifle Association but fall short of demands from survivors of last week’s school shooting.

Faced with massive protests, the Republican governor announced a plan to raise the minimum age to buy any firearm, including semiautomatic rifles, from 18 to 21. Mr. Scott also vowed to strengthen rules to keep weapons away from people who have mental health problems or injunctions against them for stalking or domestic violence.

“I want to make it virtually impossible for anyone who has mental issues to use a gun,” said Mr. Scott, who is widely expected to run for a United States Senate seat this year. “I want to make it virtually impossible for anyone who is a danger to themselves or others to use a gun.”

Student protesters who rallied at the State Capitol on Wednesday had demanded a complete ban on military-style assault rifles. On Friday, students said they were disappointed, if unsurprised, that Mr. Scott had stopped short of a ban, but vowed to keep pushing for one.