Yes, he could have requested trial by combat, but his request would have been declined for any of several reasons.
1. King Tomnen declared it illegal.
Under the influence of the High Sparrow, King Tommen Baratheon discontinued trials by combat, deciding it is barbaric and an easy way for people of influence and power to evade justice. In doing so, he robbed his mother, Cersei Lannister, of her best chance to escape any sort of punishment from the courts as she can no longer play her trump card by using Gregor Clegane as her champion.
2. Even if the North still allowed trial by combat, he might not get support.
The North might not enforce declarations of a king they considered illegitimate, and thus still allow the practice. However, Baelish has few friends in the North, and many northern lords are wary or suspicious of him. I don't see any vassals of House Stark putting their lives (or their best soldiers' lives) on the line for him even if they did want to keep the right of trial by combat.
If Littlefinger requested a champion, any person he chose might decline. We have twice seen times when people declined to be a champion. When Tyrion requested trial by combat at his trial in the Eyrie and asked for a volunteer, none of the men in the Eyrie volunteered until Bronn did. When Tyrion requested trial by combat after he was accused of killing Joffrey, Cersei chose the Mountain, and Bronn declined this time because he did not want to fight against the Mountain. It's possible that Littlefinger being disliked in the North, nobody would fight on his behalf.
3. He was not a born into a major house.
Another factor that could prevent Littlefinger from getting his request is that trial by combat is a right given only to highborn noblemen and noblewomen. He's from a minor house that once had no lands, no army, no maester, no name, no gold, and just a small keep for a castle. Although he acquired more lands, an army, and gold through his position as Master of Coin and from his scheming, he was not born into a major house. This means he was not a highborn nobleman, and thus ineligible.