Context for the answer
Has Sacred Scripture and the Church referred to the human embryo as a child? And the answer is YES! [cf. RSVCE returned search results for 'with child' and the Didache, 2].
The OP's position as well is that the human embryo is an unborn child.
Proceeding to the question
I read two questions here
1. Does the Catholic Church consider abortion to be murder? And
2. Is abortion exactly the same as murder, under the natural moral law
and in the eyes of the Catholic Church?
Not all unlawful or unjust killings are murder
First Does the Catholic Church consider abortion to be murder?
Are both unlawful killings? And the answer is yes. If by murder is meant
the unjust killing of an innocent person, I have yet come across any official Church literature or declaration that makes that equation.
In Catechism of the Catholic Church 2271, the Church interprets the Didache portion on abortion as follows:
You shall not kill the embryo by abortion and shall not cause the
newborn to perish1
1. Distinguishing abortion from infanticide.
Thus at least from this, the gravely moral evil of killing an embryo by abortion is not equivalent to the unjust killing of an innocent person.
Second Is abortion exactly the same as murder, under the natural moral law and in the eyes of the Catholic Church?
The Church portion has been answered as above. For natural law, we go to what the fifth commandment, and it says
You shall not kill. All the killings prohibited by this commandment for example intentional homicide, abortion, euthanasia, and suicide are listed here in the Catechism of the Catholic Church.
Therefore, whilst the Church considers direct abortion to be the very grave moral evil of killing an embryo, contrary to natural law, it does not equate it to murder/intentional homicide = unjust/unlawful killing of an innocent person.
Please note CCC 2323 Because it should be treated as a person from conception, the embryo must be defended in its integrity, cared for, and healed like every other human being.
it should implying that universally it isn't treated as such.
He illogically assumes that if the Church, at certain times and in
certain circumstances, regarded abortion as less than homicide it
probably "tolerated" or even "sanctioned" abortion. This is roughly
equivalent to arguing that it must be all right to kill a privately
owned racehorse because such an act does not constitute homicide. -
c. 23rd paragraph.
Early Christian writers consistently classified abortion as a grave
evil even though they did not uniformly agree that all abortion
(particularly of the unformed fetus) is equivalent to homicide. - c.