The report was issued to observe the 44th anniversary of the Supreme Court decision.
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Donald Trump said during the presidential campaign that he would appoint "pro-life justices" to the Supreme Court.
If the court voted to overturn Roe, "it will go back to the states, and the states will then make a determination."
Based on an analysis of party control in state legislatures and governorships, as well as existing laws and constitutional provisions, the report said 22 states were at the highest risk of banning abortion. Most are in the central and southern US.
Four of them — Louisiana, Mississippi, and North and South Dakota — have laws in place that would automatically make abortion illegal if Roe v Wade were to be overturned.
Eleven states were classed as an intermediate risk. And in the remaining 17, abortion rights were said to appear secure.
"We cannot go back to the days before Roe, when some women put their lives on the line when they needed to end a pregnancy," said Nancy Northup, the Center’s president.
President Trump has pledged to nominate a replacement promptly for Justice Antonin Scalia. Any nominee who opposed the Roe decision would have the same view Scalia did.
But Trump might need to put two more appointees on the court in order to make overturning Roe a realistic possibility.
Sen. Jeff Sessions, Trump's nominee for attorney general, said in his confirmation hearing that while he opposes abortion, "It is the law of the land, it has been so established and settled for quite a long time and it deserves respect. And I would respect it and follow it."
Pete Williams is an NBC News correspondent who covers the Justice Department and the Supreme Court, based in Washington.