The problem here is one of player/DM communication.
Something I learned decades ago: when a new player makes a paladin, or any other alignment-champion kind of character, it's my duty as a DM to present him, at character creation, with a full outline of my views, and my in-game world views on his alignment, his religion, his order, his expected conduct, etc. I make sure the player knows this stuff and agrees to it. If I didn't have it pre-printed, I then type all this up and bring it to him by the next game session in a format he can keep in his character folder and refer to as needed.
Of course, he may not like these restrictions, in which case I suggest, during character creation, that there might be better class choices. In my game world, there are many orders of paladins, each with different faiths and different moral codes - maybe I suggest a different order that fits closer with his idea of his character.
But once the player has agreed to the expected rules of conduct, I hold him to them.
In the OP's example, Lawful paladins don't resort to vigilante executions. They let the local authority dish out punishements. To a lawful paladin, executing a captured criminal is a violation of lawful conduct. Good paladins don't torture anyone. If an execution is merited, swift dispensation of justice is the way to go. A slow, torturous death by dragging over two days is not good behavior, even in a society that views such methods as "execution" rather than "torture". Two days of agonizing lingering death is unaligned at best, and evil in any society that doesn't condone this as "execution".
So, if this paladin were unaligned, or evil, and belonged to an order that allowed vigilante and/or retributive exections and/or torture, then all is well.
But if he's good, he may find his methods of dispensing justice called to question for his choice of brutality and torture.
And if he's lawful, he may find his methods of abdicating proper authority's rights to dispense justice and taking maters into his own hands called into question.
In general, society's tolerance for a Dirty Harry type of law enforcement tends to be much more lenient than the tolerance of a specific organization (a church or order to which a paladin might belong). In other words, the local authorities might look the other way, and the citizens might look the other way, or even commend the paladin for ending the threat of this vile criminal, but his own order will not likely look the other way if he violated their codes of conduct. It's easy in any society for a law enforcement official to serve justice and be viewed by society as a hero, but still lose his job and his position because he crossed lines that his organization would not tolerate, hero or not.