As the other answers suggest, your science and reasoning are sound. Such a room, if spun at the right rate, could simulate gravity, and perhaps negate some of the effects of prolonged exposure to microgravity. This is also fully possible with modern technology.
But for what it actually does, no one will pay for it.
Expensive to fuel
Moving a pound of material into orbit costs on average between 2,000 and 20,000 dollars. To keep this room spinning relative to the rest of the station, you would need to constantly burn fuel (or sacrifice most of your solar input). Electricity is valuable, and fuel costs too much to move up, so such an endeavor may not be accepted.
The difference between this approach and entire stations that spin is that the "one-room" approach involves friction. The friction between the spinning room and the rest of the station will slow it down, requiring more motion to combat, and thus more fuel. Meanwhile, if you spin the entire ship, it will keep spinning unless acted upon by another force - requiring significantly less energy.
Expensive to build
As noted above, moving mass into space is expensive. I'm not going to make any assumptions about the size of this room but if you want something large I wish you luck finding an investor. It's just cheaper to move the whole station.
Not very effective
- A few hours each day will not help significantly. Spending the rest of their hours without gravity will reverse most of the positive changes you make, if not all.
- Bone density is lost in space because bones no longer need to stay strong enough to fight gravity; that material is better used elsewhere. By lying on their backs, or with their backs to the wall, bones will act as if humans are sleeping; there is not much work that they need to do, so density will not increase by much.
- Other problems, such as impaired vision, occur in space because internal fluid is not restrained by gravity. By pulling from the front of the individual to the back, you will simply change where the fluid pools - in the back - and drain it from the wrong places - as opposed to pulling it down from the brain.
- Bed sores will occur from astronauts constantly lying down (as opposed to standing up produced by upright motion).
Consider spinning the whole ship, indefinitely, and having people stand upright.