Speculating someone's Jewish isn't anti-semitism unless followed by "Jews are bad." or the like. If he wasn't Jewish, he should have shrugged it off. Proved how he isn't etc. If he was, and killed himself cause somehow that's a negative to him, good riddance.
Actually, I somewhat made that point. Bad enough that it seems to be GOP playbook point to think you can smear someone in Missouri by calling him a Jew. Worse yet becoming suicidal because people are smearing you over it.
And actually, speculating that someone is Jewish can indeed very much be anti-semitism, because as usual, it is a matter of context.