There are a lot of interesting hints about the nature of Jesus in this verse, but we have to examine what he said closely to learn anything. As a human, I do not understand the Divine so it is not my place to say what Divinity can and cannot do without other aspects of Divinity. We must start by accepting that human understanding of Divinity is on par with a cockroach’s understanding of sub-atomic physics. Our goal should be learning not proclaiming our understanding. I can claim only to understand the words here, not the nature of Divinity.

This verse was spoken in the context of Jesus defending himself against the accusations that he was claiming to be God by saying he is the Son of God. This is part of a long response that was probably a dialogue where the questions and statements of his challengers was not recorded. In another verse in another encounter of the same type, John 10:34, Jesus defends himself from this charge by quoting Psalms 82:6. This echoes the idea in Exodus 7:1 that God has made us all gods.

To get at what he actually said in this verse, let us go to the Greek so we don’t get lost in the simplifications of the English translation. The Greek is:

Οὐ δύναμαι ἐγὼ ποιεῖν ἀπ᾽ ἐμαυτοῦ οὐδέν:

A literal translation of that, word-for-word for word: “No, really, I am powerful enough myself to create nothing from out of myself.” (You can see each word defined, parsed, and explained in this article.)

Notice that this verse starts and ends with a negative and the Greek uses double negatives that don't work in English, but the message is clear especially if we think of the "nothing" as "anything" to eliminate the worst of the double negative. Jesus is telling us about the negatives, the mistakes we make in thinking about our (and his) power of judgment.

Also, notice how much the Greek text emphasizes the “self”. The “self” in this verse comes from literally from the Greek word “ego”. This verse is about ego and the power of judgment.

We learn three things from this verse, not only about Jesus’s power of judgment but our own:

1. That Jesus knew that his “self” was not God. As a human, he had an "ego" or what he referred to as a psyche (see this article), which is translated as "soul" and "life, but is more like what we refer to as "ego." Our word "ego" comes from the Greek pronoun that means "I myself." This was not what made him divine any more than our egos make us divine. Quite the opposite. It makes us human and limited.
2. He knew that his power for making judgments did not arise from his ego. This is especially true when talking about judging other people. We cannot judge others if our egos get in the way.
3. His ego was not the source of anything that he created. Creation is the ability of the Father, given to us because we are made in His image, However, that ability does not come from our egos.

Whatever the Divine is, we define it as the source of creation. We see the creative, consciousness within us as the spark of the Divine. If we see Jesus as an instantiation of the Divine, his primary role was acting as an exemplar for us.

As a historical person, he existed in a specific time and place as the center of his own reality. In other words, he had an ego, but that ego wasn’t the source of his power or his judgment. Nor are our egos the source of our power and they should never be the source of our judgments.