Tl;dr: OP probably overestimates the importance of the alleged practice of holmgang (dueling) in the judicial process or the settlement of conflict in general in Viking Age and medieval Scandinavia.

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Sorry for the really, really late response.

......during the height of Danish raiding in the west

Sorry again, I'm not sure about when OP exactly means to specify alone in this sentence.
In the late 9th century? Or, around the turn of the millennium (so-called "Late/ Second Viking Age)?

If the former is indeed the case, the situation would have been much closer to that of Iceland, discussed by /u/glasgallow and linked by /u/BookQueen13 below. It was not until the middle of the 10th century that now Denmark saw somewhat stable, unified (and supra-local) polity in form of Jelling dynasty after the collapse of the Godfred-Horik dynasty around the middle of the 9th century.

On the other hand, if OP ('s supposed war veteran) lived under the reigns of the Jelling rulers like Svend Forkbeard or Cnut the Great, he (as a war veteran, I primarily regard the Dane in question as "him" in order not to go into the details on the woman's legal capacity in Viking Age and Medieval Danish laws) might get interfered by the superior authority beyond the local society level like these kings.

Another possibly (and most probably) problematic premise in OP's question is the assumption that he [the Dane ex-Viking] could use press his claim by resorting almost solely to the practice of holmgang with ease there either in the late 9th century or in the 10/11th centuries.

The study of conflict settlement in medieval Europe (including Scandinavia and Iceland) has advanced greatly especially since 1970s, with help of the anthropological approach as well as feud study, initiated early in the 20th century by Otto Brunner. From their point of view, the use of violence and the judicial process were not dichotomous (mutually exclusive), but often complemented each other to reach an agreement between both parties involved with - that now we regard as an primary goal of a series of conflicts.

As for the social meaning of not unlimited/ ritual use of violence in early medieval society (though not in Viking Age Scandinavia in a narrow sense), I also recommend you to check /u/EndOfTheWorldGuy's recent post in: Was the need to go to great lengths to defend ones honor as vain as we perceive it today in our modern times? How important was it to gain and protect honor besides the obvious reasons of ego?.

Another trend since 1970s that has affected the alleged importance of holmgang, or the legal text recorded later on medieval parchments in general, is a revisionism in the character of medieval Scandinavian law books (codes). These law texts had often been regarded as a residue (or at least including residue layer) of the Viking Age society and sometimes used for the comparative analysis to reconstruct the distant past, but now more and more scholars got used to analyze them primarily as the product of a social change in later times, especially the High Middle Ages when these texts were codified (, though a few scholars has countered this revisionist trend further now).

The majority of study on the holmgang had been published until 1970s - thus belonged to the older (now dated) approach of conflict settlement as well as the legal culture of the Viking Age Scandinavia. In other words, less and less scholars are now not so sure about the possibility that Viking Age Scandinavians actually practiced holmgang to settle the conflict widely, exact as specified in one of High Medieval Swedish lawbooks as well as some sagas of the Icelanders.

Aside from these two changing trend of research as well as the essential primary texts, however, we mainly have two more problems to take OP's case into consideration:

1. As Vogt remarks, we have not so ideal- or very limited amount of primary sources for the study on the judicial practice in Medieval Denmark. Almost no court record is extant until the end of the 14th century, and almost no record of judgement for whole the Middle Ages (prior to the Reformation in Denmark in 1536) (Vogt 2014: 185f.).
2. While medieval Danes almost certainly knew the concept of holmgang itself, we rely mainly on one specific person - Saxo Grammaticus for Viking Age Denmark. 13th century lawbooks don't explicitly refer to the holmgang as a judicial practice at all.

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Then, what Saxo (about 1200) says about the holmgang, or exactly speaking, its abolition?

Saxo mentions duels across the first half of his work (legendary pre-history part), but not so much in his latter, history (10th/ 11th century onward) part. He actually mentions its abolition twice, in somewhat contradictory manners:

1. At the time of conversion of King Svend Forkbeard by Poppo's ordeal (around the turn of the millennium) (Saxo, X-xl-4, in: Christiansen trans. 1980: 21): ".....The Danes abolished the custom of judicial duelling, and that decreed that various cases should be settled by this kind of ordeal."
2. For the second time, Saxo ascribes the total eradication of the practice of judicial combats under the reign of King Harald Hen (r. 1074/76-80), as he comments as following (Saxo XI-x-7, in: Christiansen trans. 1980: 72): "For it seems to posterity that law-disputes are better settled by swearing than by steel."

For me, these accounts suggest that Saxo certainly regarded the trial "by steel" (judicial combat) somewhat as archaic way of settlement, but we cannot know the details of former practices only based on these accounts alone.

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Provincial codes from the 13th century generally specify the testimony with oath by the respected person in the district as a means to settle the problem, and I wonder whether similar way could have also been taken before in some cases.

To give an example, The Law of Jutland (1241 CE) stipulates on the boundaries between the field as following:

"If men disagree about boundaries between fields, then men of truth of the district shall mark it with stakes or with stones and thereafter swear at that place where there is disagreement that they have done it rightly. If it is both on the boundary between fields and between districts that they disagree, then four from one district and four from the other districts, those that live closest, shall decide and swear according to what they know most truthfully they have done it right, and then make it public at the assembly. But if there is any memory that it had been sworn to before, and any man is alive of those who swore, then it shall not be sworn to again. But if there is no memory that it had been sworn to before, and the king will have the boundaries settled by riding, then it will also fully stand; however, he must not come without notice so that it is secret to some of those who are parties in the dispute. It is safer that the boundary between the fields be sworn to than ridden, the those know best who live the closest.....(The Law of Jutland, II-21, English translation is taken from: [Tamm & Vogt trans. 2016: 261f.]."

If similar system was applied to the conflict settlement for OP's case, the personal connection either with local or with the external political authority must have played an important role - since the king should also appoint "men of truth" (The Law of Jutland, II-1~4, Tamm & Vogt trans. 2016: 258). Before the royal appoint became norm, it is also likely that such a role should have been played by the local elders or possibly the chieftain (perhaps on behalf of the king).

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On the other hand, surprisingly enough, it was not until the almost end of the Middle Ages that not only the nobles, but also the peasants sometimes employ the violence as a means to push their claim against their peers (that is to say, peasant vs peasant) in medieval kingdom of Denmark. Denmark did not have a unified kingdom-wide law code like Norway, and the monopoly of violence by the kingship did not easily achieved in practice, researchars argues.

There have been some excellent scholarships on such "bondefejde", so if OP can read Danish, [Fenger 1971] can be recommended as the classic of the topic.

References:

• Tamm, Ditlev & Helle Vogt (ed. & trans.). The Danish Medieval Laws: The Laws of Scania, Zealand & Jutland. London: Routledge, 2016.
• Christiansen, Eric (trans.). Saxo Grammaticus Book X-XVI, vol. 1: Book X, XI, XII, and XIII. London: BAR International, 1980.

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• Andersen, Per. Studier i dansk proceshistorie: Tiden indtil Danske Lov 1683. København: Jurist- og Økonomiforbundets Forlag, 2010.
• Ciklamini, Marlene. “THE OLD ICELANDIC DUEL.” Scandinavian Studies 35, no. 3 (1963): 175–94. http://www.jstor.org/stable/40916463.
• Fenger, Ole. Fejde og mandebod. København: Lindthardt & Ringhof, 1971.
• ________. Gammeldansk ret. København: Centrum, 1983.
• Ingesman, Per et alii (red.). Middelalderens Danmark: Kultur og samfund fra trosskifte til reformation. København: Gad, 2001.
• Netterstrøm, Jeppe B. "Bondefejder i Danmark 1450-1650." I: Feider og fred i nordisk middelalder, red. Erik Opsahl, ss. 35-72. Oslo: Unipub, 2007.
• ________. "Feud in Late Medieval and Early Modern Denmark." In: Feud in Medieval and Early Modern Europe, ed. Jeppe B. Netterstrøm & Bjørn Poulsen, pp. 175-87. Aarhus: Aarhus UP, 2010.
• Vogt, Helle. "Danish Penal Law in the Middle Ages: Cases of Homicide and Wounding." In: New Approaches to Early Law in Scandinavia, ed. Stefan Brink & Lisa Collinson, pp. 185-200. Turnhout: Brepols, 2014.

You may be interesting in this answer from u/glasgallow. They talk about the social aspects/ repercussions to the sort of scenario you're asking about. From their answer, it seems like there wouldn't be anything to stop you per se, but you might face retaliation from the wider community.