An ethical appeal is plea to get people to do the right thing. It is a call to the conscience to urge a better, more ethical action than the current action that is taking place.
By the content of Edwards' sermon, we can gather that he believed the people were being apathetic or indifferent in their relationship with God. To correct this behavior he used many ethical appeals:
And let every one that is yet without Christ, and hanging over the pit of hell, whether they be old men and women, or middle aged, or young people, or little children, now listen to the loud calls of God's word and providence.
This is near the end (as are most of his appeals). He uses a direct command to ask people to do the right thing: turn back to God.
Therefore, let every one that is without Christ, now awake and fly from the wrath to come.
In this one of the very last statements, he again uses a commanding tone, but also envokes the emotional appeal of fear in citing God's wrath. He further uses the verbs of awake and fly figuratively as he hopes the people with spiritually awake from a moral slumber and literally flee (instead of fly) from the impending doom they will face if they do not choose to follow Christ.
Three paragraphs from the end of his sermon, he uses the persuasive appeals of emotion and ethics together in these words:
And now you have an extraordinary opportunity, a day where Christ has thrown the door of mercy wide open, and stands calling, and crying with a loud voice to poor sinners; a day where many are flocking to Him, and pressing into the kingdom of God. Many are daily coming from the east, west, north, and south; many that were very lately in the same miserable condition that you are in, are now in a happy state, with their hearts filled with love to Him who has loved them, and washed them from their sins in His own blood, and rejoicing in hope of the glory of God. How awful it is to be left behind at such a day! To see so many others feasting, while you are suffering grief and perishing! To see so many rejoicing and singing for joy of heart, while you have cause to mourn for sorrow of heart, and cry because of the apprehension of spirit! How can you rest one moment in such a condition? Aren't your souls as precious as the souls of the people in the nearby town where they are flocking from day to day to Christ?
You could also argue that there is an appeal to be like everyone else. He encourages his people here to join the bandwagon with all the other happy people. He does this after 30 minutes of tearing the people down about their sin and condemnation. In these words he uses metaphor, hyperbole, and parallelism as rhetorical devices of persuasion.