Since you used the word "liable", then it seems that you're asking in a legal sense, and my response would be - generally sadly - that you probably could be held liable. That is, you could be prosecuted for "the crime" of advocating self-harm. (This was the general case of an incident in Massachusetts which was recently adjudicated, though it didn't stem from "an argument", but a young woman who in effect dared a male who was complaining about his life to "just kill yourself" - and he did. She was prosecuted, so in that sense she was found liable. (She was found guilty, and faces a sentence of 2.5 years for involuntary manslaughter. [foxnews.com]

In a more philosophical way, though, I myself don't think that you would be morally responsible ... up to a point. That is the point at which your "statement" turns into coercion, incitement or advocacy, at which point you might be "morally responsible" if you convinced someone to do something that he had not had any intention of doing otherwise. In fact, I think that was part of what decided the Massachusetts case: the young woman was found to have crossed that line.

Not sure why anyone would need to refer to self harming in an insulting manner. The world is really fucked up these days and you never know who is on the edge, so why not play it safe?

You can make plenty of comments without telling someone to kill themselfs, but if you are being pushed to such an emotional place that you feel the need to have such an outburst that says more about you than it says about them. Could be their significant other just shot themselfs, you never know exactly what's going on in someone's life.

I would recommend developing the sort of standards in your communication so that you can stand behind your words. So if you do tell someone to shoot themselfs you better be prepared for them to take that advice seriously.

No. That would mean you couldn't say "go to Hell" and they did. Your fault? Or "go F yourself " and they did. Your fault?

I would say no.

From moral stance, it depends on mental health of that person.
If he/she is healthy person, then why would you be responsible.
If he/she does have some mental issues (and you're aware of it), then you'd be seen as an heartless asshole, or mindless asshole, or both.
So I'd advice you not to tell such thing to someone you don't actually know