The litigation process for overturning that statute will be divisive and will motivate activists on both sides (though mostly the pro-abortion activists). Things would reach a moderate level of anxiety when the case reaches the Supreme Court. All hell would break lose for the few weeks after a decision overturning Roe is announced. If this happened in September or October of this year, the GOP would indeed be in serious trouble.
But then, reality would set in. A hand full of states would vote to outlaw abortion. Many other states would vote to enact common sense restrictions. A hand full of other states would take no action at all on abortion (or would take steps to add abortion rights to their state constitutions).
The political middle would not see much difference in the world. Most of the laws limiting abortion would be sensible and moderate...the crying and shouting by the pro-abort extremists would be ugly and seem stupid. Most Americans would, within a few years, would feel at ease with the new state of affairs.
The pro-abort extremists would continue to mobilize and complain....like all the other radical lefties do. But they would become part of the background noise. The pro-life forces would disband (which would be a loss for the GOP, but not a big one since there are plenty of other social issues now to keep those people fired up).
Long term, it would be a benefit for the country as a whole, and the Dems would lose a fundraising argument...they lost the Supreme Court, Roe was overturned, but the sky did not fall.