The question of whether a victim can drop the charges or dismiss the case against the defendant is entirely dependent on whether the case is criminal or civil.
In a criminal case, the short answer is no; the victim may not choose to drop the charges. In a criminal case, the charge of domestic violence is a crime, and the state is who decides whether to charge a defendant with a crime or not. The victim does not and cannot make the decision of whether to charge the defendant or not. The process behind criminal charges is often misunderstood. Most people believe that victims of crimes issue the charges. This is incorrect. Crimes are governed by the State, and it's the State that issues criminal charges, not the victim.
Because the victim is not the individual deciding whether to charge the defendant or not with a crime, the victim cannot decide whether to later drop the criminal charges. That decision, again, is with the prosecutor’s office.
It is important to remember that even though the victim is not the individual deciding whether to charge the defendant, the victim can play an important role in the criminal proceedings and in trial. For instance, if there is a trial, the victim will likely be requested to testify against the defendant. Victims often refuse to testify, and this may be a reason for the state to drop the charges against the defendant. In California a victim can refuse to testify, but there may be slight consequences, such as fines or charges brought against you. In addition to testifying, the victim can always state their opinion of the case when asked by the judge. The judge may want the victim’s opinion when debating whether to release the defendant from jail or when contemplating the defendant’s sentence after a plea deal or trial.
Now, the opposite is true in a civil case. In a civil case, the victim is the only person that decides whether to file a lawsuit against the defendant or not. If the victim does choose to file a lawsuit, they can later settle for what is agreed upon between the parties, or they can dismiss the case entirely. There is no larger agency that makes those decisions, unlike the prosecutor in a criminal case. Additionally, a victim may choose to dismiss for any reason.