Funny story, but it's gotta be bogus.....
1) Cop radar guns are very weak - otherwise they'd fry the cop and passing motorists... ergo the plane wouldn't pick it up
1a) Cop radar only has a useful range of a mile or two so it certainly wouldn't pick up a plane out over the ocean
2) Even if it could military weapon systems do not automatically arm and fire.... especially during peacetime
3) Even if they did... the jet would have been being painted by several ATC radars that would provide much better targets
4) the sophisticated fire controls in the jet can differentiate between different types of weapon radars (ie SA-7, SA-9, Patriot, Zeus AA......) and wouldn't pick up the traffic radar as hostile... even if it could detect it.
5) the Sidewinder is a Heat-Seeker, not a Radar-Seeking Missile
6) the Sidewinder is also an Air-to-Air weapon not Air-to-Ground