The NBC News/Wall Street Journal poll found 54 percent of respondents said abortion should be legal in all or most circumstances, the highest number since the survey began asking that question 10 years ago. The 70 percent support for the ruling is the highest level since 1989.
At the same time, many states are making legal abortions harder to get, NBC reported.
In 2012, Virginia became the eighth state to require women to undergo ultrasound examinations before getting abortions, and three states passed laws requiring abortion providers to have hospital admitting privileges.
Mississippi could become the first state with no legal abortion clinic, because the only remaining one is having trouble meeting the legal requirement on hospital admitting privileges. Arkansas, North Dakota and South Dakota have only one clinic each.
"When you're the only provider in a state, you become a target," Tammi Kromenaker, director of the Red River Women's Clinic in Fargo, N.D., told MSNBC.
Abortion opponents point to polls that show most U.S. voters do not want tax dollars used for abortions.
Opponents of public funding for Planned Parenthood argue that funding for women's health services such as physical exams and contraception frees up money to pay for abortions.
The anti-abortion Susan B. Anthony List notes state and local funding for Planned Parenthood has been cut in Arizona, Florida, Indiana, Kansas, New Hampshire, New Jersey, North Carolina, Tennessee, Texas and Wisconsin.
The annual March for Life, in which tens of thousands of abortion opponents march from the National Mall to the Supreme Court, is scheduled for Friday.
Sarah Weddington, the Texas lawyer who argued for the right to abortion before the Supreme Court, is scheduled to appear Tuesday in Seattle, The Seattle Times reported. Weddington said the decision will remain the law of the land at least for the next few years, because President Barack Obama is certain to appoint pro-choice justices -- but she said there is a lot of pressure against abortions.
"The availability of a safe abortion is very much under attack," Weddington told the Times. "You have states that have a majority of people in the House or the Senate opposed to abortion who are coming up with all kinds of ways to make access impossible."
Obama released a statement Tuesday.
"On the 40th anniversary of Roe vs. Wade, we reaffirm its historic commitment to protect the health and reproductive freedom of women across this country and stand by its guiding principle: that government should not intrude on our most private family matters, and women should be able to make their own choices about their bodies and their health care," he said. "Today and every day, my administration continues our efforts to reduce unintended pregnancies, support maternal and child health, and minimize the need for abortion. On this anniversary, we recommit ourselves to supporting women and families in the choices they make and redouble our efforts to promote safe and healthy communities.
Written by the late Justice Harry Blackmun and handed down Jan. 22, 1973, the 7-2 opinion in Roe said in part, "State criminal abortion laws, like those involved here, that except from criminality only a life-saving procedure on the mother's behalf without regard to the stage of her pregnancy and other interests involved violate the due process clause of the 14th Amendment, which protects against state action the right to privacy, including a woman's qualified right to terminate her pregnancy."
Blackmun conceded a state "has legitimate interests in protecting both a pregnant woman's health and the potentiality of human life, each of which interests grows and reaches a 'compelling' point at various stages of the woman's approach to term." But an abortion, until the onset of a third trimester, "must be left to the medical judgment of the pregnant woman's attending physician."
Some, but not all, of the sweep of Roe has been chipped away in subsequent Supreme Court decisions.
In 1989, 58 percent of respondents said Roe vs. Wade should not be overturned, a percentage that has risen fairly steadily over the years. But for most of the past 10 years, at least half of respondents have said abortion should be illegal in most circumstances.
This year, 44 percent said abortion should be illegal or mostly illegal.
Bill McInturff, a Republican polling expert who did the NBC/WSJ poll, believes some of the rhetoric during last year's presidential and congressional campaigns may have shifted public opinion in favor of Roe vs. Wade. These included a remark by U.S. Rep. Todd Akin, R-Mo. -- during his campaign for the U.S. Senate -- that during a "legitimate rape" a woman's body would shut down, preventing pregnancy, a statement that may have cost the Republicans the Senate seat.
The poll surveyed 1,000 adults by telephone Jan. 12-15. The margin of error is 3.1 percentage points.
Organizers of the March for Life said on their website protesters they should rally "on the National Mall between 7th & 9th streets, the same place as last year." Marchers are then scheduled to walk up Constitution Avenue, then to the Supreme Court on Capitol Hill.
Tens of thousands of protesters have attended the rally and march each year.
National Organization for Women President Terry O'Neill planned to lead a candlelight vigil Tuesday in support of Roe vs. Wade.
Celebrity Families of 2014 [PHOTOS]