The bill, which drew the protesters to the state Capitol's Rotunda Wednesday, would also allow school districts to drop sex education and require those that keep the courses to teach only abstinence as a means of birth control, The Salt Lake Tribune reported.
The bill "takes Utah dangerously backward," said Maryann Martindale, executive director of the Alliance for a Better UTAH, which organized the rally.
Current law allows parents to keep their children out of sex-education classes or ask local school boards to offer abstinence-only education.
Brad Lancaster of Tooele said parents, not lawmakers, should decide what schools may teach.
"We're being told by lawmakers what we can and can't have our children taught," Lancaster said.
He said children need to have accurate information.
"I don't want them being taught in a locker room by a 16-year-old kid," Lancaster said.
Herbert has said he will likely make a decision next week on whether to veto the bill.
"At the end of the day, I've got to sort through all the pros and cons, and ultimately make the determination as governor what I think is in the best interest of Utah and Utah's families and is good policy for Utah going forward," Herbert said Tuesday evening.