I've included the flavor text because it tells you what's supposed to be happening - the secondary attack is a punch with your fist. However, reading the power from a crunch perspective, that's not the case. You don't have to have a hand free, and nowhere does it say that this attack doesn't use your weapon. If you have a +3 magic weapon with +3 proficiency bonus and a +5 Strength modifier, your secondary attack is (+2 +3 +3 +3 +5 = ) +16 versus AC. If it were unarmed, you would have lower proficiency (generally) and no enhancement bonus (generally).
The only crunch evidence that it's not a weapon attack is that the damage is just d6 (not [W]).
If I were trying to rule this from a flavor perspective, I would say no, you don't get any of your weapon's properties on the secondary attack because it's a punch in the face (plus you have to have a hand free to punch with). From a rules as written perspective, though, it sure looks to me as though your weapon's properties, including extra dice on a crit, count for this secondary attack.