tl;dr: Following the Dobbs v. Jackson case, this is an open question.
The Equal Protection Clause
The rights to 'life' in the Constitution are generally limiting the government's ability to take life (i.e. to apply the death penalty,) not to private parties. The 5th Amendment required that due process be followed for the federal (national) to deprive anyone of life (or liberty or property.) The 14th Amendment then extended this requirement to the state governments.
The more relevant portion of the Constitution here is the Equal Protection Clause, which is also part of the 14th Amendment:
No state shall... deny to any person within its jurisdiction the equal protection of the laws.
Murder is, of course, a criminal offense in every state. A state murder statute that only criminalized killing certain classes of persons, but not others would be impermissible under the Equal Protection Clause. For example (and this is the particular example in mind when the 14th Amendment was created in the immediate aftermath of the U.S. Civil War,) it would be illegal for a state to criminalize killing a white person, but not a black person, or for it to prescribe a lesser penalty for killing a black person than a white person. This is presumably why the Supreme Court in Roe v. Wade's majority opinion (PDF) said,
The appellee and certain amici argue that the fetus is a "person" within the language and meaning of the Fourteenth Amendment... If this suggestion of personhood
is established, the appellant's case, of course, collapses, for the fetus' right to life would then be guaranteed specifically by the Amendment.
The Court did not explicitly say here which part of the Fourteenth Amendment they were referring to, but it seems reasonable to assume that they meant the Equal Protection Clause, since that's the only one requiring states to limit the action of private citizens.
So, the question then turns on what constitutes a 'person' for purposes of the Equal Protection Clause. As the Court noted in Roe,
The Constitution does not define "person" in so many words.
That is, the term 'person' is not specifically defined in the Constitution, so it's up to the courts to determine what it was supposed to mean when interpreting a portion of the Constitution. Roe, of course, ultimately held that,
[T]he word "person," as used in the Fourteenth Amendment, does not include the unborn.
The Dobbs v. Jackson decision (PDF) declined to determine a point at which a developing human fetus becomes a 'person' for purposes of the 14th Amendment, as no such determination was needed for the case at hand,
Our opinion is not based on any view about if and when prenatal life is entitled to any of the rights enjoyed after birth.
Whether the Court might adopt some definition of 'person' that includes any class of humans not yet born in a future case is something of an open question. Dobbs does seem to squarely foreclose fetal "viability" as being that point, as that would create a vague rule that could even vary from one location to another due to difference in medical care available, but it does not appear to foreclose the question ever being decided differently from Roe's determination.
Other Possible Challenges
Aside from a potential fetal right to equal protection of life, it should be noted that a federal ban on state abortion statutes could also face legal challenges on other grounds. Exactly what those grounds might be would depend on the exact wording and reach of said hypothetical law. One of the most likely such grounds would be a ruling that the federal government simply does not have any Constitutionally-enumerated power to enact such a ban. According to the 10th Amendment,
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
In other words, if the Constitution does not delegate some power to the federal government or prohibit states from exercising that power, then any federal law prohibiting states from exercising that power would be invalid.
As such, any federal law attempting to prevent states from banning abortions would have to be enacted on the grounds of some power (or set of powers) granted to the federal government by the Constitution. The Commerce Clause would probably be the most likely candidate for a power under which Congress could make such a law, though whether it would be upheld on that grounds alone is an open question and would depend on the exact crafting of the law.