No, Jesus could not sin because He was a divine person. It is metaphysically impossible for Him to sin because to sin is to oppose the Will of God. While there is only one God, He is one God in three persons (Father, Son, and Holy Spirit). Each person of the Trinity is God but there are not three Gods, only one. This means each divine person shares the same divine intellect and divine will. What one person knows, all three know. What one wills, all three will. As the Second Person of the Holy Trinity (God the Son), Jesus is God. Therefore, it is impossible for Him to oppose His own will. Were any of the divine persons able to oppose the divine will, it would mean He would be negating Himself, which is impossible to do. Since God cannot sin (meaning He cannot go against His own will), then none of the three divine persons can sin, either.
While acknowledging that in His divinity or divine nature Jesus could not sin, some may ask whether he could sin in His humanity and human nature. The Council of Chalcedon (451) did define that Jesus had two natures (human and divine), which meant He also had two intellects and two wills. Could His human will oppose the divine will? Wasn’t His human will free? Free, yes, but so is the divine will. Free does not mean being able to deny or defy reality. You and I have a free will but we are not free to become something we are not. We are not free to become divine. We are not free to turn what is good into evil nor are we free to turn what is evil into good. We are free to choose to do good or evil, and suffer the consequences of our choices.
Jesus’ human free will, like His human rational intellect, are faculties of His human soul, united to His human body but hypostatically united to His one divine person. That means his human will and divine will are always in sync, always united and working harmoniously. This does not mean He was not tempted. The devil tempted Jesus in the desert (Matthew 4:1–11; Mark 1:9–13; Luke 4:1–14), but that was an external temptation. Jesus was not born with original sin like we were. Even though baptism washed away original sin in us, we still have the effects of it, namely concupiscence, which is the darkening of the intellect, weakening of the will, and disordering of human emotions (passions)—like the desire for pleasure, comfort, happiness, etc. Since Jesus was born without original sin, He had no concupiscence. Before the Fall, Adam and Eve had no concupiscence, either. The devil could only tempt them externally as he did in the garden (Genesis 3). The devil tried to tempt Jesus externally in the desert as well. We are tempted through our wounded human nature.
What makes us human? Is it our weaknesses? Our sins? No. Our immortal souls that have a rational intellect and a free will are what make us human and what make us created in the image and likeness of God. Those who claim that the sinlessness (also called impeccability) of Christ makes Him less human are incorrect because it is not the ability to sin that constitutes human nature. The ability to reason and to make freewill decisions is human. Animals act out of instinct and computers operate on programming. Only human beings can reason and make moral choices that are either good or evil. Jesus’ inability to sin does not diminish or negate His humanity or human nature or even His human free will. As Pope John Paul the Great once said, “Freedom is not license.” It is not the ability to do anything you want, but the opportunity to do what you ought to do—the power to do the right thing for the right reason.