No, the Dragon Magic line adds damage to the push which happens whenever an enemy hits you.
Here's the text:
Effect: Until the end of your next turn, whenever an enemy hits you with a melee attack, you push that enemy 1 square.
Dragon Magic: The enemy also takes 5 lightning damage.
I read this as being either:
Effect: Until the end of your next turn, whenever an enemy hits you with a melee attack, you push that enemy 1 square.
or (if you have Dragon Magic):
Effect: Until the end of your next turn, whenever an enemy hits you with a melee attack, you push that enemy 1 square and the enemy also takes 5 lightning damage.