According to the new survey, almost seven in 10 adults say they do not support abortion under every circumstance.
Twenty-three percent said the practice should be legal with some exceptions, while 35 percent said it should be illegal except in cases of rape, incest or to save a woman's life.
About 40 percent of respondents said abortion should be wholly legal (31 percent) or illegal (9 percent).
Groups on both sides of the debate urged commitment to their causes on Tuesday, the Roe anniversary.
"With so many women and their unborn children injured and lost to abortion, we reflect on the importance of pressing on," said Americans United for Life President Charmaine Yoest in a statement.
"We remain determined to advance a sound legal strategy and are committed to working toward a day when every person is welcomed in life and protected in law.”
The National Organization for Women (NOW) praised Roe in announcing a candlelight vigil would take place Tuesday night at the Supreme Court.
"NOW affirms that women's access to the full range of reproductive health services — including safe, legal and affordable abortion care and birth control — is integral to women's ability to participate equally in our society," said NOW President Terry O'Neill in a statement. "Roe v. Wade has helped improve women's status in the U.S. and saved countless women's lives."
The WSJ/NBC survey was conducted Jan. 12-15 and has a 3.1 percent margin of error.