Overview
My comments on this started after watching the following PragerU video, regarding The Abortion Question. As a heads up, in case you don't know, PragerU is a right-wing conservative think tank so, that being said, you might already guess what their position is on this. However, I strongly recommended watching the video, so that things are put in context.
They suggest the issue is very simple, and I disagree strongly on that point, as there is no black or white, good or evil, right or wrong answer when you consider all the intricacies of the matter. Instead, what is required is careful consideration of the different scenarios that can come up, and solutions found for each of the main types of scenarios.
The Issue of Consent
If there was consent from both parties during the sexual intercourse, the mother should not be able to have an abortion without the written consent of the father. The fetus may develop in her body, but until such time as she can remain pregnant by herself, she would also need the father's consent. This is due to the shared responsibility the two parties possess with regards to this issue.
Obviously, in case of rape abortion should ALWAYS be allowed, without the consent of the rapist (this assumes the rapist is male of course).
The Developmental Journey of the Fetus
The fetus in the initial stages of development is not the same as in the later stages. In the beginning it is just a mass of cells, no nervous system developed or anything. If abortion is done early, the only "moral grounds" you can base yourself on considering it immoral are religious.
Conversely, in the later part, it is much more akin to a newborn - and thus, the act, much more akin to murder. It is only here where you can really begin talking about objective immorality. Otherwise, even an ejaculation that does not turn into a baby is immoral, since, after all, you are killing all those cells, some of which could have become living beings.
A Possible Way Forward
The previous points suggest that there should be some rough rules on when abortion is allowed. For example, it doesn't make sense to allow abortion two weeks before birth. Conversely, there shouldn't be much of an issue if it is done in the first few weeks or up to 3-4 months (the actual time may be based on careful medical consideration and should include factors such as stage of development, risk factors etc.).
If Not Allowed, How Can the Issue of Responsibility be Solved?
If there is a state apparatus that explicitly prohibits abortion, it should also TAKE responsibility for that child (ensuring it can become an independent adult), in case the parents to not want it.
The only way to deal with this additional responsibility is by moving the responsibility from those parents, to the state (i.e: indirectly, to other tax payers). This can only be done through taxation, which is a violation of the non-aggression principle and thus, also immoral.
P.S.: I apologize for any images which are "too graphic".