By this definition, you are referring to libel.
Libel/Slander is the act of hurting or destroying someone's reputation, using unfair techniques or false statements.
While these terms are very similar, there is a key difference. To be considered "slander," the claim (or comment) must be verbal. Libel, on the other hand, needs to be written or published.
Let's look at an example of each, just to make sure it is clear.
Slander
"Timmy stole my lunch money," said Carl to his whole class. Timmy did not actually steal the money, but because of Carl's claim, nobody trusts Timmy anymore.
Libel
"Jimmy punched a girl in the face! He is an abusive guy," posted Tina, despite Jimmy not actually punching anyone. Tina types away on a blog and lies about Jimmy. Now, nobody in the entire county trusts Jimmy, and the false claim is there forever.
It is important to remember the differences between these. In this case, since we are referring to published comments, our answer is libel.