What I understand is that a stable process is one whose Xbar-R control chart is within spec. In other words, no special cause present.
But can we automatically assume from such a stable process that its output will follow normal distribution as well?
Is stability and normality one and the same?
Xbar-R chart should be compared with control limits, not specification limits. I hope, you wanted to say control limits.
If we use a variable control chart and if there is no special cause, there is a possibility of the process to follow normal distribution. But we cannot be 100% sure that it distributes normally, since we don't always use all rules i.e. to be applied to control charts. Hence normality and stability can be same and cannot. Hence better to check normality of your process (data).
You are saying the process is stable and whether it can be said that output will follow normal distribution as well?
In Xbar-R chart / any variable control chart, we take samples (which is again output of the process) and based on these we are studying behavior of the process. So, we are always speaking about normality of the process by studying output of the process (products).
Again, normality in case of SPC helps us to predict about the population using indices such as Cp/Cpk Or Pp/Ppk. Unless the process is normally distributed, we cannot properly predict about the population.