The idea of a feud as a regulated, accepted legal instrument for (at least theoretically) resolving conflicts involving individuals, families, or identity groups commonly appears throughout the Middle Ages, and throughout much of Europe. Particularly for male, aristocratic warriors, it was a socially accepted or even expected means of dealing with severe insult. Such an "insult" includes things like the death of a family member at the hands of another person, the failure of someone to hold up their end of a bargain, literal verbal insult, or any number of other slights. What constituted a feud worthy act was subject to the individual societies of the time, not to mention individuals, local politics, and perhaps even continental politics. Feuds were very diverse.

While the stereotypical example of the resolution of a feud is a duel between individual representatives of the involved parties (and that was certainly often the case!), feuds could be resolved in a number of different ways. Again, these are subject to extreme variance, given the laws of the time and location. For the primarily male, aristocratic agents in these feuds, the use of legally regulated violence would have been seen as the default means of resolution, but that doesn't mean that other methods of resolution were invalid. Keep in mind that legal feuds were regulated by the lawmakers of the times and places such feuds occurred, and keepers of the law were involved in their resolution. Some feuds, particularly later in the Middle Ages, would be resolved without violence by influence of the legal system. Others might be "settled out of court", much the same way some legal cases today are. That could mean that money changed hands, extralegal reparations were made, or even that the involved parties partook of an illegal duel outside the sight of the law. The personal and particularly judicial duel was considered important to the participants, not only on the grounds of the potential outcome, but because common belief at the time (at least on the books) is that a person would win a duel through the favour of God. Ergo, the one with the moral high ground was predetermined to be the winner under the theory of Christian medieval Europe.

Hans Talhoffer, a 15th century master-at-arms, wrote or dictated at least three separate manuscripts primarily dealing with the martial techniques one could employ in the case of a judicial duel. On top of that, however, he did leave small amounts of commentary on the law and duels within them, describing the use of violence to determine a moral victor as "wantonness". He was an advocate for wholesome Christian living, sleeping around, drinking oneself to death, and the responsible and skillful use of lethal weapons. Some of his manuscripts also appear to contain "blueprints" for odd technologies, including the likes of a medieval diving suit. He and his manuscripts represent something closer to "modernised" thinking than what is traditionally associated with medieval feuds, which might suggest the concept of a feud falling out of social favour over time -- even with those responsible for training people to win them. In Talhoffer's time (which is not far removed from Götz von Berlichingen), feuds and legal conflicts had gone through extensive regulatory processes. As a result, it was only considered proper for a legal conflict to be resolved by duel in the absence of conclusive evidence.

This style of legal resolution is likely to be fairly representative of what Götz would experience in at least some of his personal feuds. In short, it was likely that Götz slogged through at least several long, boring trials. If he dueled anyone as a result, I would have expected that to be noted more specifically -- the judicial duel was a matter of legal significance, and Götz would have dueled with the "knight treatment" if set against a legal opponent of similar social class; any judicial duels he may have fought would have probably been done with sword, spear, and possibly plate harness. While Wikipedia certainly has its limitations, any significant duels would be at the forefront of information concerning Götz if they were known about. As a result, I would guess (with heavy emphasis on that word) that Götz's feuds might have been settled with nonviolent means. However, that threatens to delve deeply into the realms of the hypothetical rather than the strictly factual, so I'll leave this answer as it is and hope that it was helpful. :)