The billboard featuring the eyes of Dr. T.J. Eckleburg is associated with the ideas of judgement and God.
Personified, the eyes are impersonal but not impartial. They function as the suggestion that there is a moral code and a moral order in the world. This moral order may be disembodied, but that does not mean it does not exist.
This suggestion is made by George Wilson toward the end of the novel when he connects the eyes on the billboard to the eyes of God.
Keeping these ideas in mind, we might expect a voice for this figurative character to be expansive, distant, absolute, and perhaps more than a little sad. Not much that passes beneath the eyes in the novel can be described as good, wholesome or motivated by goodness.
...the watchful eyes of Dr. Eckleburg keep vigil over the sad happenings of the valley of ashes.
The eyes see Myrtle killed. They see Tom come to pick up Myrtle and take her to New York. They see Tom generally abuse his position of wealth and power. They see people on the way to carouse and cavort in New York City, spending time and money like it will never run out.
There is a short-sightedness to the activities seen by the eyes. The lessons learned in the story are learned elsewhere for the most part, in the privacy of Nick's reflections.