This argument was written by Kirby Anderson, a columnist and the national director of Probe Ministries International. This critique only looks at the argument under the heading “Philosophical Arguments Against Abortion”, the rest of the article can be found here, among other places.

“A third set of arguments against abortion would be philosophical arguments. A key philosophical question is where do you draw the line? Put another way, when does a human being become a person?
The Supreme Court's decision of Roe v. Wade separated personhood from humanity. In other words, the judges argued that a developing fetus was a human (i.e., a member of the species Homo sapiens) but not a person. Since only persons are given 14th Amendment protection under the Constitution, the Court argued that abortion could be legal at certain times. This left to doctors, parents, or even other judges the responsibility of arbitrarily deciding when personhood should be awarded to human beings.
The Supreme Court's cleavage of personhood and humanity made the ethical slide down society's slippery slope inevitable. Once the Court allowed people to start drawing lines, some drew them in unexpected ways and effectively opened the door for infanticide and euthanasia.
The Court, in the tradition of previous line-drawers, opted for biological criteria in their definition of a "person" in Roe v. Wade. In the past, such criteria as implantation or quickening had been suggested. The Court chose the idea of viability and allowed for the possibility that states could outlaw abortions performed after a child was viable. But viability was an arbitrary criterion, and there was no biological reason why the line had to be drawn near the early stages of development. The line, for example, could be drawn much later.
Ethicist Paul Ramsey frequently warned that any argument for abortion could logically be also used as an argument for infanticide. As if to illustrate this, Dr. Francis Crick, of DNA fame, demonstrated that he was less concerned about the ethics of such logical extensions and proposed a more radical definition of personhood. He suggested in the British journal Nature that if "a child were considered to be legally born when two days old, it could be examined to see whether it was an 'acceptable member of human society.'" Obviously this is not only an argument for abortion; it's an argument for infanticide.
Other line-drawers have suggested a cultural criterion for personhood. Ashley Montagu, for example, stated, "A newborn baby is not truly human until he or she is molded by cultural influences later." Again, this is more than just an argument for abortion. It is also an argument for infanticide.
More recently some line-drawers have focused on a mental criterion for personhood. Dr. Joseph Fletcher argues in his book Humanhood that "Humans without some minimum of intelligence or mental capacity are not persons, no matter how many of these organs are active, no matter how spontaneous their living processes are." This is not only an argument for abortion and infanticide; it's adequate justification for euthanasia and the potential elimination of those who do not possess a certain IQ. In other writings, Joseph Fletcher suggested that an "individual" was not truly a "person" unless he has an IQ of at least 40.”

This argument can be formally paraphrased approximately as follows:
1. Roe v. Wade separated being human and being a person based on viability.
2. Viability is an arbitrary criteria.
3. Anthropologists, scientists, and medical ethicists have suggested more limited criteria for a human being a person such as cultural molding, age, and intelligence.
4. These criteria for being a person would remove 14th Amendment protection from infants and some people.
5. Arguments for legalizing abortion could be extended to justify legalizing infanticide or euthanasia.
6. Separating being human and a person, by legalizing abortion, makes a slide down the ethical slope to infanticide or euthanasia inevitable.
Conclusion: Abortion should be illegal.

Individual parts of this argument commit a variety of fallacies, for example the claim that

“The Supreme Court's cleavage of personhood and humanity made the ethical slide down society's slippery slope inevitable.”

commits the slippery slope fallacy; the author doesn’t provide sufficient evidence that there is a chain of events that lead from legalizing abortion to inevitably killing infants or euthanasia. However I think the most important fallacy, or fatal fallacy, that can be found in this argument is that of ignoratio elenchi, or missing the point, the author ignores the primary reason abortion is legal is that it was ruled in Roe v. Wade to fall under the rights of a pregnant woman as guaranteed by the Constitution. From that view, viability is not an arbitrary criteria as it changes the relation between mother and fetus significantly.

By committing this fallacy the argument could be considered dead on arrival as a refutation, however it does does provide some support for it’s conclusion as a stand alone argument so I’ve given it a rating of Critical.