The justices on Monday declined to reconsider a lower court ruling that the law violates a woman's constitutionally protected right to terminate a pregnancy before a fetus is able to survive outside the womb.
"Viability" of a fetus is generally considered to start at 24 weeks. Normal pregnancies run about 40 weeks.
Gov. Jan Brewer signed the ban into law in April 2012. Nine other states have enacted similar bans starting at 20 weeks or even earlier.
The 9th U.S. Circuit Court of Appeals said last year such bans violate a long string of Supreme Court rulings starting with the seminal Roe v. Wade decision in 1973.