Okay so. Does avenging arrow destory target creature that attacked and dealt damage to me? Or does it destroy that creature only if it(that creature) looses health?
{avenging arrow} says that dealt damage, not was dealt damage. the creature needs to have damaged something else.
Note that the damage done by the creature isn't required to be combat damage, so if it has a tap ability that dealt damage or an instant or sorcery (such as {Fall of the Hammer}) causes it to deal damage, it's a legal target.