According to Solomon, whether or not it is right to publicly accuse a person for his sake, you are better off not doing such a thing for your own sake:

What your eyes have seen
do not hastily bring into court,
for what will you do in the end,
when your neighbor puts you to shame?
Argue your case with your neighbor himself,
and do not reveal another’s secret,
lest he who hears you bring shame upon you,
and your ill repute have no end (Proverbs 25.7b-11).

You find out the hard way that Proverbs 18.17 is really true.

The one who states his case first seems right,
until the other comes and examines him.

Such defeats, at the least, involve a great deal of embarrassment. You are wise to approach the person privately first. Perhaps you have completely misunderstood him. Perhaps your idea of what constitutes a “crime” is actually backwards. If so, better to learn privately than to be shamed publicly.