If someone was in the process of kidnapping your baby and you were chasing after that person, if you have a gun on you and you use it to shoot that person and then that person dies, would you be charged with murder?
What you are charged with - murder and the appropriate degree, manslaughter, felony assault or other crime - depends on the local or federal prosecutor (and possibly a grand jury), the jurisdiction and how the prosecutor views the evidence; and if the prosecutor feels they can win a conviction against you in front of a jury for the charges.
The prosecutor will look at the evidence and try to determine your motive and what actually happened: did you want to kill the kidnapper, i.e. use deadly force to stop the crime? Did you only want to wound the kidnapper and save your baby? Did you only want to wound the kidnapper but inadvertently killed him/her? Was it possible to stop the kidnapping without using force at all? Did you first try to stop the kidnapping without deadly force and then resort to shooting the kidnapper? Did you endanger others in the area by shooting?
Those details all factor into the prosecutor's decision on which of the different degrees of murder, manslaughter, etc., to charge the parent with (or, not charge). See What Is Manslaughter? What Is Murder vs. Manslaughter? | Nolo.com.
Once the prosecutor decides on charges, they will try and win a conviction against you in front of a jury for the charges.
The prosecutor could possibly decide that the case already been tried in the court of public opinion - the person saving their baby and shooting a criminal is too sympathetic to charge with a crime - and there is no use in bringing charges and not getting a conviction from the jury.
Or, the prosecutor could decide that you were legally defending yourself and family and potentially deadly force was justified; this will greatly depend on jurisdiction, i.e. the different types of "Stand Your Ground" laws or other similar laws regarding self-defense in different states.
You may be confusing the idea of being charged with a crime and actually being of a convicted of a crime. The prosecutor can charge you with a crime and take you to court, but unless the jury convicts you on the basis of evidence, you won't be convicted.
Assuming the United States, the general rule is that Self-Defense extends to defense of others in similar situations and your immediate care. As your baby is your responsibility, you may protect the child in the same manner as any propety you may protect. There may be some issues with states that have duty to flee (which is to say, you do not have the right to invoke self-defense if flight is a valid choice) vs. Stand your ground (if you have a legal right to be in that place, you have a right to defend yourself regardless of viable flight options).
There could be some problems if the baby may have been threatened by reckless endangerment by you shooting at a moving target, holding a live human, who is fleeing, but the prosecutor is going to have one difficult case convincing twelve people at random (some of whom are definitely parents) that the shooter was more of a threat to his/her own child than the kidnapper. Consider the case of the Father who nearly assaulted the Gymnastics Team doctor who sexually molested his two daughters (as well as anyone in the room) in open court in front of the judge! From an objective and strictly legal read of the law, the judge was well within her right to throw the book at this man for contempt of court, let alone attempted assault and possible an aggravated one at that. However, given the factors motivating the man's actions, she decided not to press the charges and let him off with a very strongly worded reprimand which carries no legal weight (U.S. Judges are very territorial in their own court. Do not get one mad at you.) and I think a restriction to the back up court room where the trial was being played live on TV. She likely figured the Contempt charge (which she is judge and jury for those) and the assault charge (which, as this happened on live national tv, was widely popular... to the extent that saying the only three people who were likely to convict the man were in that court: The Doctor, who was in fear of his life, the Balif, who now has to do paperwork, and the Judge, who was more offended that it happened in her court than she was offended it happened at all.) would likely not found an unbiased panel of 12 of his peers.
It’s better to be judged by 12 than carried by 6.
To answer the title of your question: Legally whether it was a crime or not would depend upon a host of factors not mentioned, including but not limited to, the jurisdiction, where the child was, where you were, exactly what the kidnapper was doing at the time you fired, what the kidnapper had done previously, your relationship, if any, with the kidnapper, you child’s relationship, if any, with the kidnapper. Dep being upon the charge, for gen your thoughts at the time could determine whether you had legally committed a crime. So, no easy answer.
Outside of the courtroom, do your best to protect your kid and damn the consequences. Worst case scenario could make you wish you had shot your kid if that was the only way to prevent what followed. And unless the kidnapper is known to you, you have no basis for deciding which end of the scale a successful kidnapping attempt will fall on.
As to the question in the body, would you be charged: there is a vast difference between
- Committed a crime
- Arrested for committing a crime
- Charged with committing a crime
- Prosecuted for a crime
- Convicted of a crime
The police may arrest you without sufficient evidence to charge you, the prosecutor may charge you while believing there is insufficient evidence to prosecute you or may believe that there’s sufficient evidence to prosecute but that other factors make a conviction unlikely or not worth pursuing.
Prosecution is at the discretion of the DA’s office, which is either an elected position or an appointment by an elected official. And some prosecutors will be more concerned with politics than guilt or innocence, let alone justice. IMO whether you would be either initially charged or eventually prosecuted or not, would have little to do with technical guilt or innocence or the evidence for either, but how sympathetic they think you would be to the general public. They might charge you just to get a good gauge for public opinion.