Been in this business for years and I have never seen a U S Federal statute that states that the PIC of an aircraft is also a law enforcement officer. Don't recall taking that oath or getting that training. Don't confuse Part 91 of the FARs that says PIC is final authority on the safe operation of the aircraft as making the PIC a demigod or even a deputy marshall.
There are some state laws that may get into this, but if they are contrary to federal law I wouldn't want to hang my hat on them as aviation is federally regulated. I believe there may be a few state laws conferring some kind of status on PICs. Alaska has a state law that requires that a rifle, among other things, be carried as survival equipment in aircraft operated in that state. Neat trick when they go across the border to Canada.