Would Peter have gone to Caesarea if the Holy Spirit had not spoken to him?

The account indicates Peter would not have gone to Caesarea on his own, because he viewed Gentiles as unclean and outside God's dealings. Only the supernatural vision, the men at the gate, and the Holy Spirit's direct command moved him to go.

Les explains that the timing was perfect: just as the men from Caesarea arrived at Simon the Tanner's gate asking for Peter, Peter had just come through the vision. The Holy Spirit told him, "Behold, three men seek thee. Arise therefore, and get thee down and go with them, doubting nothing, for I have sent them." Les asks his classes whether Peter would have gone without such a supernatural environment—the trance, the men at the gate, the voice of the Holy Spirit speaking—and answers, "No way." Peter still regarded Gentiles as dogs, and thought God could have nothing to do with them. He was not being pig-headed; as a Jew he knew what was expected and was in accord with the program thus far, as in Matthew chapter 10.

Acts 10:19

While Peter thought on the vision, the Spirit said unto him, Behold, three men seek thee.

Would Peter have gone to Caesarea without the supernatural events?

No, supernatural intervention was necessary. The trance, the men at the gate, and the Holy Spirit’s voice compelled Peter to go. As far as Peter was concerned, Gentiles were still dogs and God could have nothing to do with them, so he needed divine intervention to visit Caesarea.