A federal appeals court last year said the restrictions were unconstitutional.
The high court’s refusal without comment to intervene now means the provisions passed in 2012 cannot be enforced.
Twelve states have similar laws, shortening the period the high court established four decades ago, after it said abortion should be legal until viability, when a fetus is generally believed to survive outside the womb. Current medical science puts that at about 24 weeks into the pregnancy.
Legal challenges in other states are working their way through the courts, including in Texas, which passed a variety of measures over access to abortion.
The case is Horne v. Isaacson (13-402).