Not at all.
People who are mentally ill are much more likely to attempt suicide then people who are not mentally ill. Most mental illness do have markers that are based in brain chemistry and affected by genetic markers. Plus, a strong indicator or symptom of mental illness is a predisposition towards self harm. For instance, bipolar disorders are often first diagnosed when a patient begins to demonstrate self-destructive tendencies.
Futhermore, many medications that would suppress symptoms of schitzophrenia are depressants - Most medicated patients describe feelings of powerlessness, impotence, disconnection and depression.
Also, the specific study that John Hopkins made examined families. Obviously, family life can play a huge role in suicide. Specifically, John Hopkins looked at families who already had histories of mental illness.
Now imagine this - you are a teenager. Your mother is being medicated for bipolar disorder and seems constantly 'out of phase'. You are not able to connect with her, or receive approval from her due to her own conditions. Your father (if he is still around, mentally ill patients have a ridiculously high incidence of divorce) is the sole breadwinner, at work and stressed. You are then diagnosed with bipolar disorder yourself and begin taking medication that makes you woozy, nauseous and makes the world feel ephemeral. In addition, your going through the typical teenaged problems (your peer group makes fun of your mental health, your boyfriend or girlfriend breaks up with you, you don't make the sports team or you get bad grades, your stressed with SATs and college applications, the economy's in a recession, etc.)
Under those conditions, it's pretty obvious that OF COURSE the subject is going to be depressed and anxious. It would actually be abnormal for you NOT to be depressed under those circumstances, because depression is the natural response to stressors, which you are receiving in abundance.
So, it's not that 'suicide' is genetic - But mental illnesses, which can be genetic, have a causative effect on suicidal tendencies, as well as creating the conditions where suicide can occur.
One of the major challenges of the modern psychological community is to combat this tendency by trying to engage the community with patients who are mentally ill and vice versa; Trying to foster a feeling of legitimate social connection and positive human interaction can be an important factor in countering a mentally ill person's feelings of despondency and giving them a sense of hope for rehabilitation.