Yes, judicial duels definitely occurred- though their popularity and conditions varied widely throughout the middle ages and in different locations. They tended to get less common as time progressed. The specific example I'll use comes from 15th century Germany (or what would later become Germany). However, the way these were done was quite different from the portrayal found in GoT. In game of thrones an accused can just demand a trial, and reap all the benefits of their physical prowess. In German society, however, care was taken to make the fight as "fair" as possible, which often meant requiring the same equipment or even purposely hindering a more capable combatant.

To see this, we can look at Hans Talhoffer's fighting manual. Talhoffer was a fighting master, and ran a successful martial arts school in the 15th century. He covers many combat situations but most interestingly he covers a judicial duel of a wife and husband fighting. This duel is not treated as anything in particularly unusual, and we can learn some things about judicial duels from his description. The man is made to sit in a hole with a wooden cudgel. The woman stands on the ground with a stone in a cloth sack. Talhoffer illustrates many ways for each combatant to win the duel, but there's a clear indication that the woman was perceived as disadvantaged and therefore was given advantages to make the fight fair.

This same dueling setup appears in other fight manuals as well, for instance in the Solothurner Fechtbuch (early 16th century)

It's significant that these duels appear in training manuals, since this might indicate that they were common enough to be thought relevant.

You can find the full fighting manual here, though I can't attest to their translation

It should be noted, however, that some scholars have argued that these pictured duels are anachronistic. Allison Coudert in "Judicial Duels Between Husbands and Wives" argues that these are historical depictions in the fighting manuals, depicting duels which occurred pre-13th century. She cites the lack of records of such duels past 1200ce.

However, she also notes that "judicial duels were common enough in the medieval and early modern period to merit etiquette books" page 27.