Your boyfriend is insecure. This is when a person assumes their other significant ones are having an external relationship when it isn't the case.
Since he broke up with you, the best option is to contact him. I would prefer text over the call. Let him know how you were faithful to the relationship. Let him know how much you love him and how much he has hurt you for something you didn't do.
Ask him to reconsider his decision/the relationship once again. To give the relationship another chance because what he had said about isn't true.
If he doesn't respond for a few days or responds negatively, exercise no contact rule.
Don't contact him because it won't be worthy to do so. Even if you reunite but still he distrusts you, the relationship will break up again. He had to gain your respect. You can't force it to him but you can show him through your words that you were never unfaithful. It's up to him to believe you or not. You can't insist too much.
After a month of silence, contact him if you still want to try. If he is still adamant not to get back with you, it's time to move on.
During the no contact, reflect whether you would like to get back with your ex. When you determine that you want to get back with him, don't be depressed when he refuses. It's an indication you need to move on.