Cycling does *stress* the neck muscles if you have a more aero riding position than an upright one.

There is an afflication that long-distance riders suffer called Shermer's Neck. It was first really documented after it affected Race Across America rider Mike Shermer. It basically involves collapse of the neck muscles so the rider cannot hold up their head.

There have been all sorts of weird and wonderful solutions to get people through to the end of long-distance events such as 1200 randonnees -- including wearing neck braces (real or made of newspapers), and holding their heads back with an inner tube wrapped around their helmets and tied down their back in some way.

If you have weak neck muscles, get some exercises specified by your doctor or physio. Bike riding won't necessarily strengthen them the way you think.

FWIW, former world Formula 1 motor racing champion Nigel Mansell suffered a broken neck in a race (I think in Formula 3 before he got to grand prix racing). Because of the extraordinary cornering forces of F1 cars, he undertook a regimen of neck-strengthening exercise to the point where he could lie on his back and do back arches with only the TOP of his head and his heels on the ground.