You can’t “get” anyone to forgive you. They have to make that choice on their own. All you can do is take responsibility for what you did and apologize, which you have done. The rest is up to him.
You don’t mention how long your guy’s been giving you the silent treatment, so it could be one of two things: either he’s cooling off because he’s still very angry, or he’s punishing you for your mistake. Some people like to pout and to rub our noses in it even after we apologize.
His silence is making you feel like a bum, but don’t buy into it. You aren’t a bad person. We all make mistakes. Some of us make enough for three or four people. You apologized, and he needs to get over it. In the meantime, be sure to let yourself off the hook, too. It’s water under the bridge and you should both move on, but as long as he’s sulking and you’re responding to his sulking, you’re both stuck in a stagnant pool of fish turds and industrial waste.
I say just leave the guy alone and let him get over it. Each time you reach out to him for absolution is just another chance for him to throw it back in your face. You’ve done everything you can do, so just back off and give him some space until he can get his panties unbunched. He’ll stop sulking if you stop acknowledging it.