There are a couple categories of "due process" which are capable of being enforced.  The two biggest are "procedural due process" and "substantive due process".

"Procedural due process" means, in its core sense, that the procedures which are used - by a government agency, court or other governmental entity - when dealing with a person (a juridical person, i.e., people, business entities, non-profit entities, anything capable of being in a court or affected by governmental action) and that person's rights are carefully spelled out, established in a neutral way, and evenhandedly applied.  Thus, a court procedure which would say "Procedure A applies in this court", but which was applied as "procedure A for white people, less favorable procedure B for non-white people" would fail to give procedural due process to B.  Similarly, Kafkaesque procedures, secret law, and such also fail to provide procedural due process.  The big question judges have to address when dealing with a question of procedural due process is "is the process fair?"

"Substantive due process" is, OTOH and in the wingnut view, more concerned with whether the result is "Fair".  Of course, the "fairness" of a "fair" result depends upon whose ox is being gored.

The difficulty which wingnuts have with substantive due process as expressed through Roe is the same one which the liberals, labor and downtrodden had with Lochner, and which the abolitionists and human rights campaigners had with Dred Scott.  In each of these cases, the losers viewed the result as illegitimate because they viewed the result as having come through the application of a powerful elite's values to determine what was "fair" and that result was contrary to what the losers sought.  If one were a slaveholder or had come up from politics in a state which was the navel of American slavery, as did Chief Justice Taney (of South Carolina) in Dred Scott, one would see a result which consigned people of color to slavery as a "fair" and "just" result.  Taney came up with many, many different arguments all of which proceeded from the passions and prejudices of the society in which he grew up.  Try reading Dred Scott some time, and you'll see a lot of different explanations, few if any of which are solidly grounded in common law.

The same can be said about Lochner.  Remember, that case was a challenge to a law governing contracts - setting maximum hours for work.  The challenge to that law was couched in terms of "freedom of contract", i.e., that people should be free to contract to do anything and the government stay out.  The justices of the Lochner court had come up through the rabidly pro-business and anti-worker society of the post-Civil War Gilded Age and were devoted to preserving and enhancing the rights of the big money to which they owed their position.  So, they came up with a lot of arguments which supported the position they favored, few if any of which were grounded in common law, and came to a result which they deemed "fair".  Again, a matter of whose ox was gored.

The same can be said, to a degree, about Roe.  There really was little said about privacy or individual autonomy of women (there had been a good bit about mens' autonomy) prior to the 1960s.  The idea of a "right to privacy" had really never been articulated as such until the turn of the 20th century when Louis Brandeis, then a lawyer in private practice, created and articulated it out of whole cloth as an argument for a family he was representing in a dispute with a media outlet that had published something the family wanted kept out of public view.  In other words, the "right to privacy" started its named existence as something akin to a "right against paparazzi".  That is not to say that there was not a right of privacy extant beforehand - one need only look at what the Founders wrote in the Fourth Amendment to see how jealously they wanted to guard privacy against government intrusion.  Rather, it can be fairly said that at the time of the Founding, privacy and autonomy free of governmental intrusion were deemed so obvious as to be not needing more definitive mention.  As such, a good argument can be (and was) made that these rights were subsumed within the Ninth Amendment's protections.

So, the wingnuts decided that Roe offended their sensibilities, but not really until after it was decided.  In the late 60s and early 70s, abortion was being legalized by legislatures repealing anti-abortion laws.  Roe finished that job and discussion - of public policy - a bit preemptively, and that was a part of the wingnut opposition.  But that opposition did not come until after the decision had been handed down.  Imagine, for a second, that the Dred Scott case had declared slavery legal, depriving legislatures of the discussion and decision on whether to legalize it, and did this absent an abolitionist movement, and that movement organized only afterward.  That's a fair analogy to how Roe landed in the consciousness of wingnut-land.

Another, and bigger, part of the wingnut opposition to Roe was that it gave the wingers something to be against and to rally their troops around.  Roe came down in 1973, after the upheavals of the women's liberation movement, the free speech movement, the hippie rebellion against the war and the Establishment, and (probably most of all) the civil rights revolution, all of which had taken place since 1960 - in 13 years.  In large part, the civil rights revolution had taken place through the Courts, not only in race relations but also in criminal procedure.  Even in 1973, it was considered gauche to be anti-black or anti-integration, and one had to admit that the criminal procedure revolutions of the Warren court had improved the justice system somewhat.  But, one could be anti-abortion with impunity.

So, it gave the wingers something where they could organize collectively all the backlashes against the developments of the 1960s.  Give me a place to stand and I can move the world, etc.

FWIW, and returning to the substantive due process issue - one can see something akin to how pure substantive due process decisions are structured in the Heller v. D.C. 2d amendment case last summer, and particularly in the diverging majority and minority opinions.  Substantive due process decisions come about most frequently when there is a constitutional right or provision which really hasn't been the subject of exposition, and there are diverging views on how far the right goes.  The controversy stays on - years after the decision.

After acknowledging that the right to privacy is constitutionally protected, she said that she thought that the citizens of the respective states should be able to vote -- to usher in their will -- as to any limits of that right. She seemed not to understand that if a right is protected by the U.S. Constitution, it applies equally to U.S. citizens without regard to their state of residence. A right that is constitutionally protected cannot be limited by the states, whether through legislation or amendments to a state's constitution. Constitutionally protected rights can only be extended by state action.  

BTD is simply right: both Roberts and Alito acknowledged the existence of a right to privacy at their confirmation hearings.  What they didn't acknowledge was a GENERAL right to privacy, in other words, a right that can mean whatever the Supreme Court wants it to mean.  It's something of a word game but that's politics.