A Song of Ice and Fire
Leniency does not come with disrespect, it requires remorse.
“Has Your Grace forgotten that we are at war with Casterly Rock? In war you kill your enemies. Didn’t your father teach you that, boy?”
“Boy?” The Greatjon dealt Rickard Karstark a buffet with a mailed fist that sent the other lord to his knees.
“Leave him!” Robb’s voice rang with command. Umber stepped back away from the captive.
Lord Karstark spit out a broken tooth. “Yes, Lord Umber, leave me to the king. He means to give me a scolding before he forgives me. That’s how he deals with treason, our King in the North.” He smiled a wet red smile. “or should I call you the King Who Lost the North, Your Grace?”
A Storm of Swords - Catelyn III
This evidently shows that Karstark has no respect for Robb, thinking of him as weak for losing the North and for going on this fools plight to "rescue" his sisters and avenge his father. Even the Greatjon is enraged by the disrespect and beats Lord Karstark.
“Pardon him, then,” urged Edmure Tully.
Robb stared at him in frank disbelief.
Under that gaze, Edmure’s face reddened. “Spare his life, I mean. I don’t like the taste of it any more than you, sire. He slew my men as well. Poor Delp had only just recovered from the wound Ser Jaime gave him. Karstark must be punished, certainly. Keep him in chains, say.”
“A hostage?” said Catelyn. It might be best...
“Yes, a hostage!” Her brother seized on her musing as agreement. “Tell the son that so long as he remains loyal, his father will not be harmed. Otherwise... we have no hope of the Freys now, not if I offered to marry all Lord Walder’s daughters and carry his litter besides. If we should lose the Karstarks as well, what hope is there?”
Robb had evidently given this though, with both his mother and uncle pleading with him, Robb had some reason to be defiant, something led him to his decision for execution and not a pardoning.
Robb later states clear and true why he had to execute the Lord.
“I know what I said, Uncle. It does not change what I must do.” The swords in his crown stood stark and black against his brow. “In battle I might have slain Tion and Willem myself, but this was no battle. They were asleep in their beds, naked and unarmed, in a cell where I put them. Rickard Karstark killed more than a Frey and a Lannister. He killed my honor. I shall deal with him at dawn.”
Robb's honour had been tainted for that Lord Karstark must be executed.
Not only that but Robb could not look weak in front of his men. Any common sellsword would've been executed for the act, he had to portray the same strength for those of greater lineage. Showing weakness would cause his army to lose faith in him. He had to remain strong and fair, he passed the sentence, and he had to swing the sword.
Game Of Thrones
In the TV Show, Rickard Karstark repeatedly insults Robb before his death. Not only did Rickard betray Robb's trust. He broke his honour. Killing two boys in their cell while they slep.
"It's treason to free your enemies! In war, you kill your enemies! Did your father not teach you that, boy!?"
Rickard to Robb Stark
Rickard clearly does not value Robb's decision. Again, leniency doesn't come with disrespect, it comes with contrition. Of which Rickard showed non.
"Leave me to the King. He wants to give me a scolding before he sets me free. That's how he deals with treason. Our King in the North. Or should I call him 'the King who lost the North'?"
Rickard to Brynden Tully
Again Rickard calls Robb a craven, saying he won't be able to kill him, mocking the King, his King. Not only that but he also gives up on Robb's cause. After the Iron Islanders had taken the North while they were in the Riverlands, claiming he "lost the North".
Rickard Karstark: "The blood of the First Men flows in my veins as much as yours, boy. I fought the Mad King for your father. I fought Joffrey for you. We are kin, Stark and Karstark."
Robb Stark: "That didn't stop you from betraying me, and it won't save you now."
Rickard Karstark: "I don't want it to save me. I want it to haunt you to the end of your days."
Again, Karstark expresses his disdain. He continues to mock Robb.
"Kill me and be cursed! You are no King of mine!"
Finally, Karstark gives up on the King entirely. He shows no respect even at his death.
Unfortunately I wasn't able to find the clip so I hope some quotes are enough, I will continue searching later.