(To wit: the US government is forbidden from searching people without a warrant signed by a judge for that particular person and situation. Creeping judicial decisions have in effect destroyed that restriction.)
While it's well within your right to refuse a search, it's within the private airline/airport rights to refuse you access to a secure area if you refuse to be "properly" screened and they can just turn you around at that point to find another means of transportation to your destination.
The private airline/airport has no say. The government, a la TSA, is the one who will not just refuse, but jail, you for opposing the unwarranted search.
If you try to get on an airplane with airline permission but without TSA search ("no thank you, I don't want to walk thru the metal detector, I don't want a pat down, I don't want to go in the weird radiation booth, I'm just going to get on the plane now"), they WILL arrest you. If you refuse the search and attempt to leave, they might not arrest you, but they will threaten criminal charges and very well may follow through on them.
Not empty threats by low-level agents. Systemic policy. Try it and show me I'm wrong.
The one exception is airplanes carrying 9 or fewer people: those are not subject to governmental interference (a la TSA). Many small airlines feature such planes precisely as a way to avoid unwarranted* search. If you find one offering airfare anywhere close to major airline prices (or the price of just driving to destination), let me know - seems they run about $1000/hr. I'd be surprised if TSA wasn't trying hard to get jurisdiction over those too.
(* - by "unwarranted" I mean "without adjudicated written approval by a court in response to presentable evidence regarding particular persons, items, locations, and circumstances"; the common "because one in a billion passengers might try to crash a plane" is inadequate.)
Nobody disputes that you must be searched to get on a plane. The question is whether or not anyone is in jail for coming to the airport, getting in line, being selected for an extra search, and saying "you know what, fuck this, I'm going home".
The FUD is that "yes, you will go to jail forever if you do that", but I don't think that has any basis in reality.
(To wit: the US government is forbidden from searching people without a warrant signed by a judge for that particular person and situation. Creeping judicial decisions have in effect destroyed that restriction.)