Can a space station the size of the ISS produce artificial gravity by rotating, or would it take a larger station? (I know the ISS was not designed for it.)
Any circular space station can produce artificial gravity by spinning on its axis. The centripetal (not centrifugal) force simulates a stronger gravitational force when rotational velocity is increased.
Note: The ISS doesn’t lack gravity, it’s just that it’s falling at the same rate the earth is curving away. Just thought that should be mentioned.
As AETBOND417 said, it depends on how fast you’re willing to spin the station. If a station has a radius of 10 meters, you need to spin it at 10 RPM. At 2.5 meters you need 20 RPM.
Of course you can’t make it arbitrarily small. I think there would be problems if the radius was comparable to our height, because then the head would feel much less ‘gravity’ than the legs. Also, things wouldn’t fall straight down.
It would be easy to build the ISS for simulated gravity. However, they don’t want any gravity on-board, simulated or otherwise-- That’s why they’re putting it up there in the first place. The ideal design, I suppose, would have spinng living quarters and a zero-g experimental section, but that would be an engineering nightmare, and might make it hard for the astronauts to acclimate when they go from one to the other every day.
Engineers know no fear. 
FWIW, there is a plan to have an artifical-G (rotating) small section of the ISS.