SALT LAKE CITY — The governor of Utah, Gary Herbert, a Republican, has approved a bill that makes Utah the first state to require doctors to give anesthesia to women having an abortion at 20 weeks of pregnancy or later. The legislation is based on the disputed premise that a fetus can feel pain at that point.

Many doctors in Utah and throughout the country are concerned that the plan could increase the health risks to women because fetal anesthesia is unnecessary. Supporters of the law say a fetus should be protected if there is even a chance it can feel pain.

“You’re telling women that they have to have something that’s going to increase their risk based on a conclusion that is not true,” said Dr. Sean Esplin of Intermountain Healthcare in Utah. He said that anesthesia or an analgesic would need to go through the woman in order to reach the fetus. Doctors could give a woman general anesthesia, which would make her unconscious and probably require a breathing tube, or a heavy dose of narcotics.

No other state has passed a law like this one, said Elizabeth Nash, a policy analyst at the Guttmacher Institute, a nonprofit abortion rights group. In 2015, Montana lawmakers passed a similar bill that would have required fetal anesthesia before surgeries, including abortions, performed at 20 or more weeks of gestation, but the state’s Democratic governor vetoed the measure.