Yeah, he'd have to let those pesky American voters decide.
States and voters can no more allow abortion than they could legalize slavery. The "send it back to the states" notion is not only bogus, but impossible. Either the woman has the right to choose, or the child in her womb has the right to life. The latter is the only basis for overturning Roe.
This is an issue of constitutional rights, no matter the pendulum's swing. Therefore, Thompson's anti-Roe/pro-choice position is untenable.