Can I stop domestic violence charges against the person who committed it against me?

In a criminal matter, the victim is the victim, not the prosecutor. The crime is against the laws of the state. You can sue civilly, and control what happens there, but it is up to the prosecutor to decide what cases to drop if a crime is committed.

Technically the prosecutor can continue without you willing to cooperate. Once a statement has been given to the police that is all the prosecutor needs technically. To be realistic even if you told the prosecutor that you want charges dropped and that you're not willing to testify, the prosecute probably won't drop the changes completely, they'll probably be more willing to reduce the charges or be more lenient when it come to sentencing recommendations. It is possible that they could drop the charges, but don't hold your breath. In Utah there has been a greater push to prosecute domestic charges even if the victim isn't willing to testify.

No, the state not the victims press criminal charges. If you want them dismissed contact the prosecutor handling the case.