One person I know of argues that abortion can be seen as a Thirteen Amendment issue -- a question of forced labor, literally. For instance, you cannot be forced to be a Good Samaritan, even giving a pint of blood to your child. I basically like the theme, especially since lack of control of fertility was a major problem with slavery. Nonetheless, one can take the argument too far -- basically, many understand "slavery" and "involuntary servitude" as basically being legal terms of art. Basically, certain ones have commonly understood meanings, even if they can be (and logically would be) used more broadly. A simple case of "well, I don't mean it that way" applies here. So, it is not quite an "obvious" argument, let us say.
This might be a case of strict construction, therefore, perhaps mixed with some sort of original understanding (abortion as illegal in 1868). The latter aspect is a bit harder to crack, though I think we can use it against the other side to some extend, especially by taking a historical approach -- what are the evils of slavery? Nonetheless, I think the problem can be attacked in a different way as well. Originally, Radical Republicans especially, wished to interpret the Thirteen Amendment broadly. Blacks were full citizens now, therefore they deserved full and equal rights of citizenship, including those found in the Bill of Rights. For instance, some argued the Privileges and Immunities Clause of Art. IV now covered all citizens in the nation -- states could not get around things by saying that their own citizens (or rather, blacks) are deprived of protections. Certain fundamental rights applied to all.
Some, including some moderates, were uncomfortable with this approach, especially since it meant limiting the discretion of certain Northern states. The Thirteenth Amendment itself was controversial -- it barely passed in Congress, one remember without members from the South (and loyalist Maryland already passed an abolition law), and only then with the votes of lame duck members. The problem was the second section which gave Congress the right to enforce the amendment, a possibly open-ended grant of national power that threatened traditional federalist themes.
Thus, civil rights laws that went beyond simply protecting free labor (and sometimes even there) were quite controversial. President Johnson's veto of such legislation actually was not as controversial as some might think -- clear abuses arguably were necessary for moderates to support all aspects of some of these laws. And, even then, they were a bit iffy on the constitutionality of them, while some were worried that future congresses (imagine if the Dems got back in power! horrors!) or presidents might overturn them. The arguably iffy constitutionality would add fodder to such attempts. Therefore, Congress decided to clear things up by passing the Fourteenth Amendment.
It protected the rights of freemen -- especially but not limited to newly freed slaves -- while giving Congress clear enforcement powers (listening Supreme Court?) to safeguard such rights. And, the federal courts would secure them as well, especially if local political forces were unfriendly. The amendment is an open-ended affair with vague and glorious phrases securing "privileges and immunities of citizenship," "due process of law," and "equal protection." The Supreme Court early on suggested it was mainly concerned with newly freed slaves, though soon enough its reach was deemed to be much broader.
But, the amendment clearly is specifically concerned with the evils of slavery. It was passed to guard against them, and not in a narrow and stingy fashion -- again, the amendment was ratified to guard against arguments that merely freedom from the lash of slavery was at stake. Truly, all "badges" of slavery were covered as well, and more. This is where abortion pops up again ... even if a narrow understanding of freedom of servitude would not include having control over one's body in this fashion, the more liberal reach of the Fourteenth Amendment adds much more force to the argument. The Thirteenth Amendment is strong enough -- even Chapter 13 of bankruptcy (voluntary payment plan) cannot be forced on a debtor -- but taken together, the two amendments surely secure the right at hand.
Equal protection and due process of law (including guarding against arbitrary/selective burdens, which is a glaring problem with morals prosecutions) only strengthens the claim, again, which is the point: the three prong protections of the Fourteenth Amendment along with the clear grant of congressional authority to secure them was meant as redundant safeguards to freedom. Again, a stunted view of freedom violates the very intent of the amendment, not even going into its spirit. The theme goes beyond this one area, which is why it is so important -- abortion fits into a broader range of freedoms, especially basic controls of family life and one's own bodily integrity/health.
Thus, I am supportive of the Thirteenth Amendment argument, but would take a more inclusive approach. This after all is what the Constitution invites in any number of areas -- when dealing with serious risks of harm, redundant safeguards are often in place to guard against injury. The same applies here -- thus separation of powers and checks and balances both protect us from governmental abuse/error. Religious freedom includes aspects of free expression, guards against establishment, free exercise, and equal protection. And, the "right of privacy" is but one formulation of basic liberties that are guarded any number of ways, even if just one arguably might be enough.
So no need -- actually it is a bit risky -- to rely on one specific argument. A baseball team runs into problems when it has only one weapon. Our liberty is the same way.