On the other hand, the legal system could require evidence of assault. Given the time between usually unreported assaults and the time when a woman learns she is pregnant, any evidence could be difficult to come by.
Highlighting is my own. Wanted to point out that the problem here is that the claim of difficulty in procuring evidence is wrong. Forensics science revolves around the golden rule known as Loccard's Exchange Principal, which holds that any time two things interact with one another, they will exchange elements of the interaction, no matter how small or fleeting, and without these exchanged elements, we must assume that the exchange never occurred.
A Pregnant woman who is seeking a Sexual Assault Abortion will usually have some evidence of the assault occurring. There are many terms for the most common type of evidence found, but for simplicity's sake, lets call this evidence the BABY that is inside of her.
Again, as you asked, I will not for the purposes this answer, discuss when it can be defined as human or has obtained personhood and nothing here should be construed as doing so. However, there is no documented medical evidence of a pregnancy that did not involve a male human. Everyone has come to the realization that, sexaul reproduction means that your parents had sex at least 1 time in their life time... and you're the evidence of that.
Having children is genetic: If your parents did not have kids, you aren't likely to be having children either. So the fact that a woman is pregnant means that there is enough genetic material of the fathers that still exists inside of her. The child is going to have a 50% identical match to exactly one male, which is good enough to go find the guy and ask a few question about the night they got together as well as search for eyewitnesses and ask people who are looking. It's pretty easy to find the man who is the daddy, since we have the baby and thus half of his DNA. Even if a genetic match is not 100%, we are able to tell how distantly two people by DNA... such that a match of 0.01% will likely be a 6th cousin. In a large familial extension, out to sixth cousin it's likely that a stronger match is going to be found as the chances that a person with a criminal record or having a career in teaching, law enforcement or military are likely to be identified in a CODIS system. Even submitting to DNA family tree company could trigger a CODIS hit, as they do share DNA sampling with law enforcement. It's not an uncommon news story to hear of a cold case being solved because a DNA family tree company submitted DNA that matched with an unidentified suspect DNA sample added by investigators. (Fun fact: At least 1/4 of all people on Earth are genetically related to Ghengis Khan and only one President of the United States, Martin Van Buran, was not a direct descendant of Prince John of England... the same one Robin Hood fights. Even then, that's direct descendants. Van Buran still counts an ancestor of Prince John among his relatives, meaning all U.S. Presidents, even Obama, have a connection to the British Monarchy... and we know this because DNA).
Since a parent and child have 50% of DNA in common, it's pretty easy to find the dad in such a case.
Furthermore, as none of the abortions in these laws criminlize the mother for getting an illicit abortion, the standard of proof could be considerably lower than Beyond Reasonable Doubt. Again, for academic purposes, I will not discuss how low, but stating that this standard is only used in criminal cases and thus we don't happen to be absolutely certain she's telling the truth, but probably mostly certain.