First - I’m not any of the parties involved. There ARE no parties involved, this is totally hypothetical.
Let’s say I own a business, including a commercial building. Because of vandalism in the area, I install cameras that record what they see, and I have signs clearly posted that the cameras are there. “If you chose to be on my property, you will be recorded.”
One day, a customer gets pissed off about something, and out of the blue, attacks and beats another customer, until the victim is required to go to the hospital for stitchs and a possible concussion. For some reason, the victim declines to press charges against their attacker.
Can I, as the property owner, with the video to back me up, press charges? Something illegal was done on my property, it just wasn’t done to me.
The crime was against the state. A person cannot refuse to press charges they can only refuse to cooperate at the time. The state can bring charges at its own discretion*. You as a business owner in the community can hand over your evidence and ask the prosecutor to press charges and that might hold some sway.
You could certainly file a police report and provide them with a copy of the video. In most jurisdictions (all in the U.S.) with which I’m familiar, though, it is the prosecutor or district attorney who actually presses charges, NOT private citizens. The decision isn’t yours to make. A case with an unwilling or uncooperative victim isn’t likely to be a high priority for the D.A., but they could decide to proceed, depending on workload, how badly they want the accused off the streets, etc. (this does happen rather frequently with domestic abuse cases, for example).
What slash2k said. Prosecutors decide whether to prosecute based on the seriousness of the charge, the evidence available, and the likelihood of securing a conviction. Indivduals may urge the prosecutor to charge someon (or not to charge someone) in a particular case, but unless they are unusually influential individuals this is not going to be a big factor in the prosecutor’s calculation. In the hypothetical outlined in the OP, the desire of the property owner that charges be brought is likely to be outweighed by the reluctance of the victim to testify.