Graham's bill has virtually no chance of passing the current Congress, where Democrats hold slim majorities in the House and Senate.
Some Republicans, including GOP candidates in pivotal Senate races, have backed Graham's new proposal.
But other top Republicans either refused to back Graham's bill or expressed a belief that individual states should set their own abortion laws.
"With regard to abortion, Democrats are clearly focused on abortion," he added.
As Graham's bill brought a renewed focus to the issue, Democrats pounced.