No. In the United States, it's easy to argue that the daughter, as a candidate's child, is a public figure, especially if she had appeared in any capacity on her father's campaign or spoke about her father in a political manner. This is the proverbial "Entering the Arena" that could hold her to public commentary. A good question as to whether or not she counts as a public figure is asking "But for the public remarks made by Candidate A, would the the daughter be unknown from the public or out of the media spotlight? Usually it would be a "N". As a general unwritten rule of campaigns children are off limits, but it's not a crime to disparage a politicians kids... just a public outrage.
Furthermore, in your second paragraph, Candidate A is citing prevailing rumors/reports that this did happen, so Mr. A did not make this up on his own, but repeated something from some other source. Since the rumor is already out there, it's something of public concern, which is not defamation. Candidate A could further mitigate the suit if he promptly apologized upon learning of the screw up and corrected the record. Either way, since he was speaking on a publicly available source (even if incorrect) his repeating of incorrect facts that at the time he believed were factual and relevent to the campaign.
In one particular way to get around defimation is to couch the terms in notions of other people's words. It is defamatory to say she had an abortion (a lie, she did not, and is injurious to her character) and a completely different thing to say "I have heard rumors that she had an abortion" (which is true). Whether or not "She had an abortion" is factual or false, it is factual that there are rumors, and it is factual that he has heard some rumors. He's not commenting on the verasity of the rumors, but the mere fact that the rumors exist and he is aware of them... but perfectly self-demonstrating: He just demonstrated the rumor's nature, thus proving the rumor exists, and he has heard about it (If it was started by connecting it to dirty politics, daughter has to prove that). Incidentally, this is why when we arrest someone who committed a mass murder spree that was captured on numerous security cameras and news footage, the news will refer to the individual as the "Accused Mass Murder First Second Third-Name" and not "Mass Murder First Second Third-Name", because under the eyes of the law, he's not-guilty until he pleas his guilt or is found guilty by the jury. If the News reports that he did it before a trial, (even if we watch the whole thing unfold on national tv) they can be sued by the accused because it's not yet true that he was guilty of mass murder.