Originally Published: 03 MAY 23 11:09 ET

By Tina Burnside, LJ Spaet and Shawna Mizelle, CNN

(CNN) -- Republican lawmakers in the North Carolina General Assembly have reached a deal on legislation that would place restrictions on most abortions after 12 weeks.

Assuming the agreement holds, Democratic Gov. Roy Cooper of North Carolina, who has worked to enshrine abortion rights, would be unable to block the legislation due to a veto-proof GOP supermajority in the state that was reached last month after a Charlotte-area Democrat switched parties.

Senate Bill 20, dubbed the "Care for Women, Children and Families Act," bans any licensed physician from performing surgical abortions after the 12th week of pregnancy. It provides exceptions in the case of rape and incest through 20 weeks of pregnancy or in the event of a "life-limiting anomaly" through 24 weeks.

Under current state law, abortions are legal up to 20 weeks of pregnancy. A reduction to 12 weeks would be significant but less restrictive than some other GOP-led states that have recently enacted six-week or near-total bans on the procedure.

The bill would additionally prohibit any health care provider who objects to abortion "on moral, ethical, or religious grounds" from being required to participate in medical procedures that would result in an abortion.

"The refusal of a physician, nurse, or health care provider to perform or participate in these medical procedures shall not be a basis for damages for the refusal or for any disciplinary or any other recriminatory action against the physician, nurse, or health care provider," it states.

It also calls for in-person examination by physicians seeking to prescribe or administer abortion-inducing drugs and prohibits people within the state from mailing such drugs to a pregnant woman. This comes as a legal battle over mifepristone, one drug used in many nonsurgical abortions, looms over the US Supreme Court.

The House is expected to take up a final vote on the bill Wednesday and the Senate is expected to vote on Thursday, according to CNN affiliate WTVD.

North Carolina has been a haven of sorts for women seeking abortion care as southern states tightened restrictions after the Supreme Court sent the issue of abortion rights back to the states. The anti-abortion lobbying group Susan B. Anthony Pro-Life America praised the legislative deal, saying it will "lessen abortion tourism in the state."

Democratic leaders across the state have spoken out in opposition of the bill. In a statement to WTVD, US Rep. Wiley Nickel of North Carolina said, "It's unfortunate that once again, the Republican majority in Raleigh is pushing extreme restrictions on abortion and working to criminalize women's health care decisions."

US Rep. Deborah Ross of North Carolina, a Democrat, said in a tweeted statement that the bill was "extreme and out of touch with the needs of women and the will of the people" of North Carolina.