There is no third party standing in the way as is often rumored. You are the union. No, it is not the responsibility of unions to judge whether a nurse is good or bad. Unions only ensure that proper procedures are followed. Those procedures are negotiated in your contract and voted on by you.
Most contracts provide for both written and verbal counseling.
Some require further efforts at rehabilitation or support depending on the circumstances. Hospital administrators have a right and a responsibility to discharge nurses under certain conditions. However, they don't have a right to harass employees into quitting or to fire employees without just cause. Well, yes and no. As a union, regardless which union or local you belong to, your participation is the union's strength.