Yes, public protest is the right way to seek justice as it is generally a peaceful means of making the will of the populace known. When put under enough peaceful pressure, governments tend to cave to the will of the populace. Thus, protests are a great way to seek justice.
Sometimes, the injustice is that something that happened isn't even being spoken of, acknowledged, or addressed in any way whatsoever. Public protest increases awareness about issues that have, either intentionally or coincidentally, slipped through the grasp of the media the first time around. While public protest does not always directly lead to justice, there are some scenarios that public protest gives justice a chance, where staying quiet would have accomplished nothing.
Nonviolent public protest is an acceptable and in the right instances extremely effective way of demanding and leading to justice.