Over at the National Pro-Life Alliance, there is a list of legislation that they endorse to reduce abortions in America. Two of those bills effectively overturn Roe v. Wade, and both are completely Constitutional in my view.
One is the Life At Conception Act. It's sponsors are Rand Paul in the Senate and Jim Jordan in the House.
The Supreme Court invited a definition of personhood in their ruling when they stated:
"If this suggestion of personhood is established, the appellant's case [i.e. "Roe" who sought the abortion], of course, collapses, for the fetus' right to life is then guaranteed specifically by the [14th] Amendment."
The Life At Conception Act would do just that, by asserting equal protection under the 14th amendment for all life, with life beginning at the moment of fertilization.
Congress could also take the approach long advocated by Rep. Ron Paul.
The Sanctity of Life Act would take advantage of the constitutional mechanism known as jurisdiction stripping. Article 3, Section 2 explains:
The judicial power shall extend to all cases, in law and equity, arising under this Constitution, the laws of the United States, and treaties made, or which shall be made, under their authority;--to all cases affecting ambassadors, other public ministers and consuls;--to all cases of admiralty and maritime jurisdiction;--to controversies to which the United States shall be a party;--to controversies between two or more states;--between a state and citizens of another state;--between citizens of different states;--between citizens of the same state claiming lands under grants of different states, and between a state, or the citizens thereof, and foreign states, citizens or subjects.
In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction. In all the other cases before mentioned, the Supreme Court shall have appellate jurisdiction, both as to law and fact, with such exceptions, and under such regulations as the Congress shall make.
The trial of all crimes, except in cases of impeachment, shall be by jury; and such trial shall be held in the state where the said crimes shall have been committed; but when not committed within any state, the trial shall be at such place or places as the Congress may by law have directed.
Therefore, Congress could limit the appellate jurisidiction of the Supreme Court by disallowing them to hear any cases involving abortion, essentially repealing Roe v. Wade retroactively. Abortion could then be outlawed by the states again.
With these two solutions on the table, would the Republican party act if they had the majority?
They didn't in the Bush years, and the current leadership is of the same blood. They are afraid of what the media will say, and consequently, what the voters will do.
When the next Congress begins, if the Republicans hold the majority in both Houses, new leadership will be needed if we are to overturn Roe v. Wade.
The reason many Republicans only support "appointing pro-life justices to overturn Roe v. Wade" and "a constitutional amendment to ban abortion" are because they will never happen but they can still placate Republican voters with those statements.