WASHINGTON—Top Democrats said they plan to make protecting women’s access to abortion a centerpiece of the fall elections while Republicans railed against a rare breach at the Supreme Court, after a leaked draft opinion indicated that conservatives could be on the brink of victory in their long fight to overturn Roe v. Wade.

The draft Monday evening by Politico, which it said was written by Justice Samuel Alito and was the opinion of the court. The authenticity of the draft, dated from February, was confirmed by Chief Justice John Roberts, but he said it wasn’t necessarily the final resolution in the case. The Supreme Court’s spokeswoman declined to comment.