Logistical issues with only allowing abortions for rape victims:

Problem 1: Criminal investigations and trials take forever. Even if things go relatively quickly, it can still take six months to a year before a case is heard. This means we have two options: allow for later-term (read: possibly post-viability) abortions, or let women who claim a rape/incest exception have the termination immediately and do the trial later.

Problem 1A: Most people tend to not be in favor of post-viability abortions. Ignoring the issue of whether aborting a viable fetus is ethical, late-term abortion procedures pose significantly greater health risks to the woman, are also very expensive, and tend to be illegal in much of the US.

Problem 1B: If we allow a woman who claims to be pregnant as a result of a rape to get the abortion before it goes to trial, there's nothing preventing her from refusing to cooperate after the procedure, thereby ensuring the case will fall apart and rendering the spirit of the exception pointless.

Problem 2: This policy will provide women with significant incentive to lie about being sexually assaulted. This is bad for everyone: it's bad for the women who feel like they must lie about sexual assault in order to end their pregnancies, it's bad for the men who they would be falsely accusing, and it's also bad for real rape victims, who aren't exactly taken seriously as it is. We already have "Oh, she just regretted it the next day" and "She was wearing a short skirt and led him on" and "She shouldn't have drank so much"-- do we REALLY need to add "She got pregnant and just wanted an abortion" to the list?