How Democrats and Republicans explained the Roe fallout on Sunday talk shows.

On the first weekend after the Supreme Court overturned nearly five decades of constitutional abortion rights, Democrats seized on the ruling to portray their Republican opponents as threats to women and their health care providers, while two sitting G.O.P. governors welcomed the decision, as they tried to emphasize that the matter is a local issue with more “debate” to be had.

Stacey Abrams, the Democratic nominee for governor in Georgia who is in a rematch with the Republican she narrowly lost to four years ago, told CNN’s “State of the Union,” that the public should “take into very real consideration the danger Brian Kemp poses to the life and welfare of women in this state.”

Ms. Abrams also told CNN that Mr. Kemp “intends to adds incest and rape as prohibitions.”

Tate Mitchell, a spokeswoman for Mr. Kemp, said in a statement that Ms. Abrams “is lying” and that Mr. Kemp supported the state’s law that includes exemptions for rape, incest, life of the mother, and ectopic pregnancies.

Ms. Abrams also appeared on “Fox News Sunday,” and said, “We cannot cherry-pick when we pay attention to the lives and safety of women.”

After noting Mr. Kemp refused to expand Medicaid in Georgia, Ms. Abrams said, “He has refused to support women at every stage of their lives when they are trying to make the best choices for themselves and their families.”

The CNN anchor Jake Tapper said Mr. Kemp had been invited to appear on the show. Ms. Mitchell said Mr. Kemp was unable to appear because he was at the Georgia Municipal Association conference in Savannah.

Gov. Gretchen Whitmer, Democrat of Michigan, told CBS’s “Face the Nation” that state lawmakers had already introduced legislation to “criminalize and throw nurses and doctors in jail” if they perform abortions.

And legislators, she said, endorsed a 1931 law making abortions in the state a felony “as have all of the Republican people running for governor. They want abortion to be a felony: no exception for rape or incest. That’s the kind of Legislature that I’m working with. That’s the kind of matchup I’m going to have this fall.”

Republican governors on the Sunday shows, while welcoming the court’s ruling, repeatedly emphasized that the debate and discussion around this issue will continue, framing it as a matter of states’ rights.

Gov. Asa Hutchinson of Arkansas said on NBC’s “Meet the Press” that although the ruling was something the “pro-life movement worked for over 40 years” to achieve, “we have to remember, this not a nationwide ban on abortion. Every state will have the ability to make its decisions.”

Later, Mr. Hutchinson tried to assuage concerns that other rights could be rolled back: “This is not about contraception. This is not about same-sex marriage: a very limited decision on this particular issue of abortion.”

And it is “very important right now to assure women that the access to contraception is going to be able to continue.” Later, when asked if, as president, he would sign a national law outlawing abortion, Mr. Hutchinson, who is considering a run in 2024, said no.

“I don’t believe that we ought to go back to saying there ought to be a national law that’s passed. We fought for 50 years to have this return to the states. We’ve won that battle. It’s back to the states. Let’s let it be resolved there.”

Another Republican, Gov. Kristi Noem of South Dakota, told ABC’s “This Week” that the Supreme Court ruling was “wonderful news,” and that her state would now ban abortions except to save the life of the mother. “But I anticipate there’ll be more debate and discussion” because the ruling “gave the authority back to the states to make these decisions.”

When asked what would happen if a South Dakota resident traveled to another state to get an abortion, Ms. Noem replied, “That certainly isn’t addressed in our statute today and so I think that’s things that there’ll be debate about but also, we’re having lots of debates in South Dakota.”