Can you be put in a psych ward for self injury or for suicidal thoughts?

In short, yes.: If you are suicidal, all states have provisions for emergency hospitalization for you to get the help you need.

Hospitalization: Many people who have uncontrollable desires to self-harm or kill themselves find relief in being hospitalized. Such feelings can sometimes be extremely intense. Inpatient treatment can allow focused help in a protected environment that they don't have at home. Addressing painful issues can begin, with round-the-clock help. Discharge to ongoing outpatient treatment is possible afterwards.