No amount of social justice can cure someone's schizophrenia. Some treatment is available to homeless people, but again the vast majority are not willing to participate (the drugs have very unpleasant effects, so this is unfortunate but you can empathize with someone not wanting to experience those side effects).
If someone becomes an alcoholic, loses a job because of it, becomes alienated from their family and social support network, and ends up deciding, under the influence of that addiction, that they prefer to drink at the expense of everything else, even housing, what can social justice do to help them?
There are ways to improve the situation. Decriminalization of drugs, more research into how to treat people with mental health issues and addictions, more resources smartly applied to treatment.
TLDR; People don't become homeless because of poverty or a regressive tax code, they become and stay destitute because of mental illness and addiction. Finding ways to intervene is important, but the existence of homeless people does not imply social injustice. There may be social injustice in the US, but this is not a symptom of it or proof of it.
References:
http://www.psychosocial.com/policy/dualdx1.html
http://bjp.rcpsych.org/content/162/3/314.short
http://onlinelibrary.wiley.com/doi/10.1034/j.1600-0447.2001....
http://barrettfoundation.org/causes-of-homelessness/
Quoting from that last link:
> In addition, approximately half of people experiencing homelessness suffer from mental health issues. At a given point in time, 45 percent of homeless report indicators of mental health problems during the past year, and 57 percent report having had a mental health problem during their lifetime. About 25 percent of the homelessness population has serious mental illness, including such diagnoses as chronic depression, bipolar disorder, schizophrenia, schizoaffective disorders, and severe personality disorders.
> Substance use is also prevalent among homeless populations. In a 1996 survey, 46 percent of the homeless respondents had an alcohol use problem during the past year, and 62 percent had an alcohol use problem at some point in their lifetime. Thirty-eight percent had a problem with drug use during the past year, and 58 percent had a drug use problem during their lifetime.