I have been reading this thread avidly but not commenting, until now. Mr. Burlew's article is inciteful and generally correct, and I think it lends itself well to the problem at hand (as apparently did Janx). I personally, don't see anything wrong with what the paladin did as I read it. No decision had been reached by the group concerning the fate of the orc, so the paladin did not violate his word or the word of a party member. In barbaric ogre/orc controlled lands, there is no "legitimate authority" of a humanoid sort (no king/lord/magistrate) to take the problem to, and Tyr certainly outranks them anyway, so his paladin does as well by extension. So, no disrepect shown for legitimate authority. You "could" make the arguement that running down a defenseless, unarmed orc on horse-back and slaughtering him was less than honorable, but I don't know that it qualifies as dishonorable.
However, you do clearly have a problem. This sort of head-butting, if it is causing problems OOCly as it clearly is, is bad for your game and the enjoyment of all your players. I think one of the fundamental problems you may be encountering is that the paladin is your only non-chaotic character. The paladin is a delicate class that needs the right surroundings to flourish. It needs a DM who communicates to the player directly and upfront his own views of the paladin class and is willing to work to incorporate the player's view and be consistent with what they agree on. It needs a competent player who can handle the sometimes onerous burden of the paladin's alignment and code-of-conduct without putting undue strain on the other players, and it also needs other party-members that can handle and adapt to the paladin's often unique outlook. Lacking the proper DM, the paladin is just a frustrating trap for some poor player who will constantly be harrassed for his IC behavior every time it doesn't conform to whatever unspecified opinion the DM has about the class. Lacking the proper player, you get a paladin that disrupts the game with his holy-boy antics and creates resentment amongst the rest of the group. Lacking the right party mix, you get a paladin who is just waiting to get backed into a moral quandry with no escape. I think what you have is the wrong party mix. That's not intended to be a slight on you, the player of the paladin, or any of your other players. It's just that the character concepts you've all come up with don't mesh in a way that's conducive to the paladin, and quite honestly your character leaving the group will do little to salvage it. Next time, it will just be the CG barbarian or rogue.
My suggestion is to talk to the paladin's player privately and express your concern and ask if he/she would be comfortable discussing it with the rest of the group. You do all need to hash this out, but you don't want to risk your friendship by making the player feel like you ambushed him or tried to turn the whole group against him. Perhaps just playing a paladin of a different diety would improve things dramatically. All paladins have a proverbial stick up their backsides, but paladins of Tyr tend to opt for the triple-extra-large redwood tree. A paladin of Ilmater might suit better, and a halfling paladin of Yondalla would probably have spared the orc after the confession about his children if he extracted a vow by the orc to forswear evil and teach his children not to act that way either. Or perhaps just a cleric of a diety more likely to get along with a bunch of chaotic adventurers. Ultimately, though, I think the paladin acted correctly and that continued correct action will just lead to further group strife in and out of character. It may just be time to rethink the character.