Do you have to report all car accidents to your insurance company?

Yes, auto insurance policy states that you are required to report all accidents (losses) immediately.
There are a few reasons for this.

The carrier wants to quickly see your vehicle - take photos and write an estimate. If it was a single vehicle loss, and no monies are owed to any other party, they STILL want to do this because, essentially, they are not still insuring the 'same' vehicle. It is now damaged and likely has a different estimated value. Even though damage appraisers can distinguish old damage from new, your carrier has the right to know the condition of this insured vehicle. EVEN IF you don't carry first party collsion coverage on this vehicle, you could have another accident in the future and if we view this vehicle for THAT accident, we already know the damage incurred in the prior accident.

If you were in a two car or multiple car collision, your carrier needs to speak with you as soon as possible after the accident - and you should WANT to also; you certainly want both your carrier AND the carriers of any other involved vehicles to hear your side of the story. If you don't, they have only the version given by the other parties and have to make their liability decisions based on what they've heard.