I know he has a different religion, but he could have demanded a trial through combat, why not?
Well, there never was a trial, since Eddard confessed to save his daughter -- so trial by combat could never happen.
Here are the relevant quotes from the books. It starts with Varys visiting Ned in his cell:
Varys tsked. “Cersei will not want to hear that, I promise you. Stannis may win the throne, but only your rotting head will remain to cheer unless you guard that tongue of yours. Sansa begged so sweetly, it would be a shame if you threw it all away. You are being given your life back, if you’ll take it. Cersei is no fool. She knows a tame wolf is of more use than a dead one.”
“You want me to serve the woman who murdered my king, butchered my men, and crippled my son?” Ned’s voice was thick with disbelief.
“I want you to serve the realm,” Varys said. “Tell the queen that you will confess your vile treason, command your son to lay down his sword, and proclaim Joffrey as the true heir. Offer to denounce Stannis and Renly as faithless usurpers. Our green-eyed lioness knows you are a man of honor. If you will give her the peace she needs and the time to deal with Stannis, and pledge to carry her secret to your grave, I believe she will allow you to take the black and live out the rest of your days on the Wall, with your brother and that baseborn son of yours.”
Ned isn't very forthcoming, so Varys reminds him that his family is also at risk:
“So what is your answer, Lord Eddard? Give me your word that you’ll tell the queen what she wants to hear when she comes calling.”
“If I did, my word would be as hollow as an empty suit of armor. My life is not so precious to me as that.”
“Pity.” The eunuch stood. “And your daughter’s life, my lord? How precious is that?”
A chill pierced Ned’s heart.
“My daughter …”
“Surely you did not think I’d forgotten about your sweet innocent, my lord? The queen most certainly has not.”
“No,” Ned pleaded, his voice cracking.
“Varys, gods have mercy, do as you like with me, but leave my daughter out of your schemes. Sansa’s no more than a child.”
Varys gave a long weary sigh, the sigh of a man who carried all the sadness of the world in a sack upon his shoulders. “The High Septon once told me that as we sin, so do we suffer. If that’s true, Lord Eddard, tell me... why is it always the innocents who suffer most, when you high lords play your game of thrones? Ponder it, if you would, while you wait upon the queen. And spare a thought for this as well: The next visitor who calls on you could bring you bread and cheese and the milk of the poppy for your pain … or he could bring you Sansa’s head. The choice, my dear lord Hand, is entirely yours.”
The next time Eddard appears in the book is when Arya sees him confess in public:
Her father raised his voice and began again. “I am Eddard Stark, Lord of Winterfell and Hand of the King,” he said more loudly, his voice carrying across the plaza, “and I come before you to confess my treason in the sight of gods and men.”
Her father raised his voice still higher, straining to be heard. “I betrayed the faith of my king and the trust of my friend, Robert,” he shouted. “I swore to defend and protect his children, yet before his blood was cold, I plotted to depose and murder his son and seize the throne for myself. Let the High Septon and Baelor the Beloved and the Seven bear witness to the truth of what I say: Joffrey Baratheon is the one true heir to the Iron Throne, and by the grace of all the gods, Lord of the Seven Kingdoms and Protector of the Realm.”
All that's left is the sentencing:
The High Septon knelt before Joffrey and his mother. “As we sin, so do we suffer,” he intoned, in a deep swelling voice much louder than Father’s. “This man has confessed his crimes in the sight of gods and men, here in this holy place.” Rainbows danced around his head as he lifted his hands in entreaty. “The gods are just, yet Blessed Baelor taught us that they are also merciful. What shall be done with this traitor, Your Grace?”
Prince Joffrey … no, King Joffrey … stepped out from behind the shields of his Kingsguard. “My mother bids me let Lord Eddard take the black, and Lady Sansa has begged mercy for her father.” He looked straight at Sansa then, and smiled, and for a moment Arya thought that the gods had heard her prayer, until Joffrey turned back to the crowd and said, “But they have the soft hearts of women. So long as I am your king, treason shall never go unpunished. Ser Ilyn, bring me his head!”
Ned Stark was offered a plea bargain where he would plead guilty, be sent to the Night's Watch, his son would be Lord of Winterfell, and his daughters would come to no harm. Joffrey, like the inbred mentally challenged idiot that he is, decided to execute Ned against the wishes of the small council.