The measure is among new limits on abortion pushed by Republicans in several states, some of which have prompted court challenges.
Existing Florida law forbids abortion after 24 weeks' gestation, unless a woman's life or health is jeopardized by continued pregnancy.
The new law, effective next month, sets the no-abortion point at any stage of development when a doctor determines the fetus is viable.
Once fetal viability is attained, no abortion would be allowed under the new law unless two doctors certify in writing that termination is necessary to save the patient's life or to "avert a serious risk of substantial and irreversible physical impairment of a major bodily function ... other than a psychological condition."
The bill's sponsors argued modern medicine has made it possible to save infants far earlier than the third-trimester standard for pregnancy termination used by the U.S. Supreme Court in its 1973 ruling that legalized abortion nationwide.
Critics accused Scott and Republican legislative leaders of failing to respect women's rights.
"They don't think women should be able to control their own health care decisions," said Barbara DeVane, a Tallahassee lobbyist for the National Organization for Women (NOW).
DeVane said she did not know if NOW would take the issue to court in Florida, but predicted patients will challenge the law.