Can this sliding door hardware support solid wood doors?
As long as it is a solid wood door and not a veneer. Drill the proper sized pilot holes too, so the door rail does not split.

The only thing you may want to be concerned about the screws are is the one on each roller that will go into the end grain of the stile. If it was a longer screw, it would be good insurance. End grain does not hold a screw as well as the grain running perpendicular to the screw. There, a 2 1/2"-3" screw would be well placed. Running it at a slight angle help cross the grain a little too, making it grip better. That trick will not be needed on the other screw.