Before you take any step towards him, you need to go deeper into your values and check on what ground you want to stand in life. Is lying okay? cheating? etc.
The truth is that it happens a lot in life. I am not sure where you stand with that and your ex has the same question. You apologize but you are not actually taking a definite decision for this not to happen again. He knows that and in his mind, he thinks: "It happened once... It will happen again..."
Relationships are usually based on some form of commitment.
I believe it is your choice.
It is your call.
You are the one who decides where you want to stand with that.
To tell you the truth, another guy might go with it. Your ex however is very clear: he won't play the game based on these rules.
If you want a chance with him, you'll definitely have to shift your mind set.