Keeping silence fallacy?

It's not quite the same thing, as you have informed them that you aren't going to engage with them due to their behaviour, but Argument from Silence is pretty close.  I think it's really just another ad hominem, as your refusal to have further discussion has no bearing on the truth of someone's claims.

Argument from Silence

Implying that you are some kind of globe model agent looks to me like an ad hominem (circumstantial) suggesting you are biased or predisposed to take a particular stance.

Letting them know you will no longer discuss anything with them as they had resorted to an ad hominem, they have ignored your clear reason for 'silence' and instead done an avoiding the issue fallacy by not addressing your point.

Sounds like an argument from ignorance to me.  We know because we don’t know.  Silence is negative proof.  Silence is the absence of information.  You can’t know something from nothing.  Think of Miranda rights.  You have the right to remain silent.  

Is a kind of Argument From Silence Fallacy, and a kind of False a Dilemma Fallacy as in either you are with us or with mainstream media and its also a kind of Guild By Association Fallacy, or a Position The Well Fallacy and a Scapegoat Fallacy as they are using that to smear you. 

I don't accept the default of "silence is compliance", particularly when the compliance is with the opponent's view.

It is another of the limitations of pure logic.  The default of "silence is consent" was widely asserted, but did not help Thomas Becket when the "powers that be" (Henry II) wanted to imprison/behead him.  Logic is rarely the deciding factor in national issues.