Can you be blamed for someone's suicide?
The short answer to the question of whether you might be liable (legally "blamed") for someone's suicide is "no," because there are only special, limited cases (e.g., having legal obligation to take care of the person; providing the means used for the suicide; engaging in a deliberate campaign of harassment to cause it) that might result in liability.
The question asks whether you could be "blamed" for another's suicide, but the relevant question is whether you could be "liable," or held legally responsible, for it. (After all, anyone could be "blamed" for anything; the real question is, does "blaming" you have any consequences?)
While there is a longer answer below, the short answer is "no". Absent the special circumstances discussed below, one person is not responsible for another person's choices and actions, including suicide, and so in the vast majority of cases, there would be no liability for another's suicide.
The law is full of exceptions, and the following exceptions, while narrow and often self-evident are real. They represent situations where person 1 could possibly be legally liable for person 2's suicide:
(1) If there was a special relationship involving an enhanced duty of care between you and the person who committed suicide, such that you were required to look out for them but failed to do so, that could create liability. The most common examples are teachers or administrators in schools, who might be liable (or at least their schools may be liable) for failing to stop bullying and harassment leading to suicide, or counselors/therapists who essentially commit malpractice by failing to spot any warning signs of a patient's impending suicide, if a reasonable counselor or therapist should have done so.
(2) If you assisted in the suicide, including by giving pills, a firearm, etc., to someone whom you knew wanted them in order to kill him/herself.
(3) Related to the second one -- if you were unreasonably careless in leaving the mechanism (or means for suicide) around someone whom you reasonably should have known posed a suicide risk. For example, leaving a loaded gun, or gun plus ammunition, accessible in a household where there was a despondent person, since doing so could be seen as negligent, or unreasonably careless.
(4) If you engaged in a pattern of harassment or bullying with the intention, or at least the reasonably logically foreseeable outcome, of causing suicide. You may have read of recent cases in the news, where there is an attempt to hold someone liable for not just bullying another person but actively encouraging him or her to commit suicide. Deliberately attempting to cause the suicide is what may lead to liability.
The law in this area is evolving, so the exact outlines of liability for another's suicide are not yet known. But the basis rule is, most people--including most family members, significant others, friends, co-workers, etc.--will not be liable, since again, one person is not responsible for another's choices or actions. There must be something more to possibly impose liability, such as wrongful or careless act, or special relationship imposing a duty under the law.