With what right would he have executed the guy ?
Leaving aside the fact that he doesn't want to stand out which he defiantly would if he just randomly executed a noble, who also happens to be a seemingly famous adventurer, he has no authority to punish him. That authority lies with the country or the guild or whatever, he would only get in trouble with the country for being some random upstart that decided he had the right to judge nobles and adventurers and execute them according to his own whims.
It would be a different thing if they were in battle and killing him would protect people but in this case where there is no imminent danger it's far more reasonable to just leave the criminals judgment to the people who's job it is to do so, the people that officially have the authority to do so.
He could use his connections to go the long way and have the guy arrested and tried for his crime, where I doubt he will get anywhere close to executed for attempted murder, you know being a noble and famous adventurer, he probably gets away with a fine or maybe prison time, and possibly a demotion in his adventurer rank or getting kicked out of the guild, but that's a whole lot of effort and wasted time for something the MC doesn't really care for, especially considering the guy is broken enough and there are enough witnesses that it will happen eventually either way.